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University  of  California  •  Berkeley 

The  Theodore  P.  Hill  Collection 

of 
Early  American  Mathematics  Books 


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■'■^^E 


'  ^^ift  %B9i  ^a^  ysbtM     S^aS^ 


USINESS  CALCULATOR 


>AND(  

: ACCOUNTANTS  ASSISTANT' 


A  CYCLOPEDIA 

OF  THB 

Most  Concise 

AND 

Practical  Metlioiis 

OF 

Business  Calcinations. 

INCLUDINO  MANY  VALUABLE  LABOS-SAVINQ  TABLES, 


TOGETHER   WITH 


Improved   Interest  Tables, 

DECIMAL     SYSTEM : 

SHOWING   THK   INTEREST   ON   FROM 

$io    to    $10,000 — Rate,    Ten    per    Cent,    per     Annum. 

BY 

HOY  D.  ORTON,  and  W.  H.  SADLER, 

President  and  Founder  of  Sadler"- 

"  Bryant  &   Stratton"  Business 

College,  Baltimore,  Md. 


Lightning  Calculator,  formerly  teach- 
er of  Rapid  Calculations  at  the 
U.    S.  Naval  Academy. 

Designed /or  the  practical  use  of  the 

Banker,  Merchant^  Accounta  »■?,  Mechanic^  Farmer^ 

Business  Man   and  Student.      Contaic^tng  the  shortest^  simplest  ana 

most  rapid  methods  0/  Computing-  Numbers,  adapted 

to  all  kinds  0/ business  and  every-day  life. 

Written  and  arranged  so  ?is  to  be  within  the  comprehension  of  every  one 

having  the  slightest  knowledge  of  figures. 


PRICE     ei.OO. 
Sent  to  any  part  of  the  World  on  receipt  of  same» 


BALTIMORE.  MD,: 
W.  H.  Sadler,  Publisher,   Nos.  6   and   8  North   Charles   Street 


Entered  according  to  Act  of  Congress,  in  the  year  ^877,  by 
In  the  Office  of  the  Librarian  of  Congrees,  at  Washington. 


The  pagoB  of  reprint  from  "Orton's  Liohtntno  Calculator"  are 
protected  by  copyrights  of  Hoy  D.  Orton,  issued  in  1866  and  1871. 


N.  B.—AU  rights  reserved.    Any  infringement  roill  be  prosecuted  to 
the  fullest  txtent  of  the  law. 


OENERAL    AGENTS. 

p.  0.  Address,  Nos.  6  &  8  N.  Charles  St,  Baltimore. 

JAMES  G.  MOULTON,  WILLIAM  CALLEN,  Jr., 

JOHN  G.  SCOUTEN,  WILLIAM  DAVID. 


AGENTS  A^TANTED.-For  particu- 
lars  and  territory,  apply  in  person  to  either  of  the  above 
General  Agents',  or  address  the  Publisher. 

ORDERS. 

Parties  ordering  the  "  Calculator  "  should  be  par- 
ticular to  write  plainly  their  Name,  Residence,  County  and 

Suae. 

Upon  receipt  of  One  Dollar,  a  copy  of  the  book  will 
be  forwarded,  jx)st-paid,  to  any  address. 

W.  H.  SADLER,  PflMer, 

Nos.  6  &  8  N.  Charles  St., 

BALTIMORE,  MD. 


^-= 


The  principal  features  embodied  in  this  work  are  sim« 
plicity  and  brevity. 

There  can  be  nothing  new  in  principle,  but  so  far  as  the 
authors'  knowledge  extends,  their  peculiar  methods  and 
abbreviations  in  the  practical  applications  of  the  rules  of 
Addition,  Multiplication,  Fractions,  Percentage,  Interest, 
Averaging  Accounts,  and  Mensuration,  have  not  hereto- 
fore been  published,  except  such  as  are  contained  in  -the 
work  of  the  senior  author,  known  as  "  Orton's  Lightning 
Calculator."  As  an  endorsement  of  Professor  Orton's 
original  work  on  the  subject  of  rapid  or  lightning  calcula- 
tions, it  may  be  here  stated  that  over  400,000  copies  of  that 
book  have  been  sold.  It  is  not  the  design  of  the  authors 
of  this  work  to  make  a  text-book  for  the  use  of  beginners 
in  arithmetic,  but  to  offer  to  those  who  have  mastered  the 
principles  of  addition,  subtraction,  multiplication  and 
division,  a  guide  to  the  practical  application  of  that  knowl- 
edge of  arithmetic  and  calculations  which  is  required 
daily  in  business  and  the  affairs  of  life.  There  is  no  quali- 
fication more  essential  to  success  than  facility  in  the  rapid 
and  accurate  use  of  figures. 

In  view  of  the  increasing  demand  for  the  original  work, 
which  has  been  several  times  revised,  the  present  authors 
have  decided  to  enlarge  upon  the  subject.  They  present 
in  this  new  volume— the  result  of  their  joint  labors— the 
most  extensive  and  comprehensive  work  of  the  kind 
ever  offered  to  the  public,  in  the  full  assurance  that  who- 
ever will  carefully  study  its  pages  will  glean  therefrom  aa 
abundant  reward. 

Baltimore.  Jidy,  1877. 

3 


-d^!^ 


CONTENTS 


Prefacr 3 

Frontispikce — Illustration ...  4 

Introduction 9 

Addition 12 

*'        Lightuing  Method 13 

—Table U 

"        Illustration 15 

An  Easy  Way  to  Add  17  to  26 

Multiplication — lUuatrated 26 

Short  Methods 26"  31 

**                  Coutractions 31  "  33 

**                          "        •   — Curious  and  Useful 33 

"                  T.^l)le  of  Squares 34 

Fractions 35 

"          !M<'ntal  Operations 36 

"         Wli.n  tlu'  Sinn  of  the  Fractions  is  One 38  "  41 

"         M  lien  till'  Fractions  have  a  like  Denominator. ...  41 

*'          lxai)iil  Procf'ss  of  Multiplying  Mixed  Numbers 41"  48 

"         ^Vhea  the  Multiplier  is  an  Aliqiiot  Part  of  100. . .  48 

♦♦          Tableof  Aliquot  Parts  of  100  to  1,000 48 

"         Counting-Room  Exorcises 49  "  53 

"         Illustration 53 

"         Division,  with  Analysis 53  '*  55 

«                 '         by  Boxing  55,  66 

"          Multiplication  aud  Division 56"  59 

Percentagi;— As  apjjlled  to  Business 69 

"             Illustration 60 

♦*             Given  Cost  and  Selling  Price  to  find  the  Rate  62 

"             Given  Profit  and  Rate  to  find  the  Cost 63 

•*             Given  Amount  and  Rate  to  find  the  Cost  ....  63 
•'             Given  Proceeds,  showing  Loss  and  Rate,   to 

find  the  Cost 64 

Profit  and  Loss— Illustration   65 

"                 Short  Business  Methods 66 

*•                  Table  of  Aliquot  Parts 66 

IhTERESi— Showing  Application  of  Percentage 67  "  72 

5 


6  CONTENTS. 

Discount— Commercial 73 

True 74 

Bank 76 

Commission — Illustration 77 

'  Siiowing  Application  of  Percentage 77  to  80 

Insurance — Showinu;  Application  of  Percentage SO 

Investments — Illustration 81 

"  Capital  and  Stocks,  showing  Application  of 

Percentage 81  "    85 

"  Table  for  Investors 86 

Intfrf-st  Discount  and  Average — Illustration 87 

"  Simplifietl  by  Cancellation 93 

"  Short  Practical  Rules 100 

Banks  and  Banking— Illustration 104 

Interest — Bankers'  Method 105  '*  114 

Lightning  Method 114  "  117 

Merchants'  Method 117  "  124 

Partial  Pa  vments — Notes,  Bonds  and  Mortgages 124,     129 

Equation  of  Pavmenis 129,     135 

Averaging  Accounts — Illustration 135 

•'  Lightning  Method 135  "  139 

Partnerships  13d 

'  Settlements  by  Three  Diifferent  Methods 140  "  144 

Gold  to  Currency— Gold  at  a  Premium 144 

CuHRENCY  TO  Gold —    "        "        "  144,     145 

Maturing  Notks,  etc 145,     146 

Sterling  Exchange— Illustration 147 

How  Calculated 148    '  151 

"  OldTable 151 

"  New  Method  and  Tables 152,     154 

Harking  Goods— Illustration 154 

"  Asking  Price  and  Discounts 155,     159 

*♦  Rapid  Process 159,     162 

"  Table  for  Marking  all  Goods  Purchased  by 

the  Dozen 162 

Basis  of  Success  in  Business 163 

Ledger  Accounts— Illustration,  or  the  Science  of  Book-keep- 
ing Comprised  in  a  Few  Pages. . .  .164  "  175 

How  TO  Close  the  Ledger 175,     177 

Balancing  Bcok.s — Illustration 176 

Errors  in  Trial  Balances — Illustration 178 

How  to  Detect  Them 178  "  l^'6 

Lumber  Measuring — Illustration 1^3 

"  Short,  Practical  Rules 183    '  li'ti 

Measuring  Coed  Wood— Illustration IbG 

Short,  Practical  Rules 186  "  189 

RoundTimber— Measuring— Illustration ISW 

•'  Short,  Practical  Rules 189  "  193 

Flooring —  **  '  "      193 

Square  Timber — Measuiing — Illustration 194 

Short,  Practical  Rules 194  "  196 

Cisterns  and  Reservoirs— Table  <  f  Capacities l'.*6 

*'  Illustration 196 

•*  Iluw  to  Measure  their  Contents.  197  "  2U0 


CONTENTS.  7 

Cask  Gauoino— Illustration 200 

fSl'orr.  I»ractical  Rules £(»3 

Mkasurino  Geain— Jlliistratiou 203 

"                  Si/o  of  Bins,  IIow  Ascertained 204 

*                   Weights  and  Measures,  U  S.  Standard .  204 
"Wkiohts  and  Measures  >  Tablo  of  Avoirdupois  Weij^hts,  and 

BrsHEi.s  TO  Pounds          j  No.  of  Pounds  to  the  Busliel 205 

Ir*)N  Weights — Used  in  Railroading — Table  of  Estimates.. .  206 

Cons  JN  CuiBS — Measuring — Illustration 207 

'•  J'raetical  Rules  for  Estimates 207  "  210 

Measuuino  Hai— Ilhistratiou. 

*'  Estimating  Quantity  in  Stacks,  Mows  and 

Meadows 210  "  213 

Weight  of  Live  Catile— Illustration. 

*'                      "Weights  Estimated  by  Measure- 
ment   214 

Builders'  MEASUREMENXS-Illustration 215 

"                        Bricklaying 216 

Tiling  or  Slating 217 

Walling 218 

Ghi/ing 220 

"                        Plumbing 221 

«                        Masonry 222 

"                        Plastering 223 

Short  Rules  foh  the  Mechanic — Illustration — 225 

Square  and  Cube  Roots 226  "  237 

Mensuration,  or  I'ractical  Geometry 2U7  "  246 

Tabus  or  Multiples 246 

AvoiRDUPOis  Weight— Illustration,  with  Tables 247 

''                     Ljngor  Iron  Ton 247 

*'                     Iron  and  Lead 247 

«                      Miscellaneous  Table 248 

Apothecaries'  Weight— Illustration,  with  Tables 248 

'               Ki.uiD  Measure 249 

Dry  Measure — Illustration  and  Tables 250 

Cubic  or  Solid  Measure — Illustration,  with  Tallies 251 

Measurements — Valuable  Information  Concerning 252 

Nails — Sizes  and  Number  to  the  Pound 252 

Liquid  Measure — Illustration,  with  Tables 253 

3Ieasurements — Linear  ob  Long — Illustration,  with  Tables  254 

Surveyors'  Measure 255 

Geographical  and  Astronomical  Calculations — Tnblo. . .  255 

Surface  or  Square  Measure— Illustration,  with  Tubks. . .  256 

Surveyors'  Square  Measure,  with  Tables 257 

Contents  of  Fielps  and  Lots — Table 258 

Fencing — Tablo  showing  the  No.  of  Stakes,  Rails  and  Posts 

Required  in  Fencing 258 

Troy  Weight — Illustration,  with  Tables 269 

Diamond  Weight , 259 

Paper,  Books  AiND  Stationery— Illustration,  with  viuiuus 

Tables 260 


O  CONTENTS. 

PfiiNTiNo — Typfi-eetting 261 

'*          Press-work 262 

"          CosttfPaper  268 

"         Table  showing  Cost  cf  Paper  by  the  Quire 264 

Books— Sizes  and  Styles 262 

Shoemakers' Measure 263 

Measurement  OF  Time— Illustration 265 

Table  266 

"               "               Circular  Measure 266 

LoNOiTCDE  AND  Time — Table 267 

Time — How  to  Ascertain  the  Difference  between  Cities 267 

Table — For  Ascertaining  the  No.  c  f  Days  between  Two  Dates  268 
•'         Showing  the  Number  of  Days  from  any  Day  in  one 

Month  to  the  same  Day  in  Another 269 

ASTUONOMICAL  CALCULATIONS 270  tO  274 

MoN  e V  of  tJie  U  nited  States— Illustration 274 

"         Fiance 275 

"         the  German  Empire 276 

Arbitration  op  Exchange 277 

Value  of  Foreign  Coins  in  U.  S.  Gold 278,  279 

Gold  and  Currency  Values 280 

Bank  Accounis — Illustration — How    to  Transact   Business 

with  Banks 281  "  284 

Interfst — Commercial  Rules , 284,  285 

Interest  Tables— Decimal  System , .  ,286  "  290 

"             Compound  Interest ..  295 

Time  Required  for  Monet  at  Interest  lo  Doublk 296 

U.  S.  Interest  RATts  and  Penalties , 297,  298 

How  to  OnrAiN  Wealth 299 

Wages— Value  of  Time 300 

Ready  Reckoning 301 

'•                 •'           Tables 302,  303 

Taule  of  Illustuations 304 


i  fs/^^J  G^^l^il^n^rr-*^ 


w^Sai^lli^ 


,^r^ 


Quantity  is  that  which  can   be  increased  oi 
diminished  by  augments  or  abatements  of  homo 
geneous  parts.      Quantities  are  of  two  essential 
kinds,  Geometrical  and  Physical. 

1.  Geometrical  quantities  are  those  which  occupy 
space ;  as  line$^  surfaces^  solidsj  liquids,  gases,  etc. 

2.  Physical  quantities  are  those  which  exist  in 
the  time,  but  occupy  no  space ;  they  are  known  by 
their  character  and  action  upon  geometrical  quan- 
tities, as  attraction^  light,  heat,  electricity  and  mag- 
netism, colors,  force,  power,  etc. 

To  obtain  the  magnitude  of  a  quantity  we  com- 
pare it  with  a  part  of  the  same ;  this  part  is  im- 
printed m  our  mind  as  a  unit,  by  which  the  whole 
is  measured  and  conceived.  No  quantity  can  be 
measured  by  a  quantity  of  another  kind,  but  any 
quantity  can  be  compared  with  any  other  quantity, 
and  by  such  comparison  arises  what  we  call  calcit' 
lation  or  Mathematics^ 


10   ORTON  &  Sadler's  calculator. 

Introduction^. 

Arithmetic  means  reckoning  by  numbers,  calculating. 

Notation  means  writing  numbers. 

Numeration  means  reading  numbers. 

Number  is  one  or  more  things  or  units,  as  one,  two,  &c. 

Unit  or  one  is  a  single  thing. 

Numbers  are  represented  by  figures. 

Figures  are  characters  used  in  Arithmetic  to  represent 

numbers. 
All  numbers  are  represented  by  the  ten  following  figures: 

(WrUten)      <^  /.    J.    J.    J.    J".     ^  /.    cf.  f. 
Cipher,  one.  two.  three,  four.  five.  six.  scYen.  eight,  nine. 
(Printed)       0.      1.     2.      3.      4.      5.     6.      7.      8.      9. 
These  figures,  except  the  cipher,  are  often  called  Digits. 
Digit  means  the  measure  of  a  finger's  breadth. 
Figures  were  called  digits  from  counting  the  fingers  in 
reckoning. 
The  character  0  is  called  a  cipher,  from  the  Arabic  word 
tsphara,  which  signifies  a  blank  or  void.     The  uses  of  this 
character  in  numeration  are  ko  important,  that  its  name, 
cipher,  has  been  extended  to  the  whole  art  of  Arithmetic, 
which  has  been  culled  to  cipJier,  meaning  to  work  with 
figures. 


INTRODUCTION.  11 

Standard  Measures,  to  prevent  error  are  generalhf 
derived /rom  nature.  For  example,  measures  of  time. 
from  the  time  of  the  revolution  of  the  earth  about  itf 
axis :  of  space,  from  the  length  of  a  barley-corn,  taken 
from  the  middle  of  a  full-grown  ear ;  also,  from  th« 
circumference  of  the  earth  ;  of  weight,  from  the  weigM 
of  a  grain  of  wheat,  taken  as  above  ;  also, /rom  th0 
weight  of  a  definite  quantity  of  distilled  water ;  of 
heat, /rom  the  temperature  of  boiling  water,  &c. 

The  four  principal  operations  of  Arithmetic  are 
represented  by  the  following  signs : 

-f  Plus  or  more,  the  sign  of  Addition 
—  Minus  or  less,  "        Subtraction. 

X    Fnto  (multiplied  by)  "        Multiplication. 
-J-  By  (divided  by)         "        Division. 

When,  in  solving  a  question,  only  one  operation  if 
need,  the  answer  has  a  distinctive  name. 

In  addition,  the  answer  is  called  the  sum. 

Subtraction,  ••  "I  Difference  or 

(  Remainder 

Multiplication,       '*  "  Product. 

Division,  "  "  Quotient. 

A  sign  made  thus  =,  called  Equal  to  or  Equals,  is 
placed  between  two  quantities  to  show  their  equality ; 
Thus,  1  +  1  =  2  is  read,  one  plus  one,  equal  to  two  ; 
or,  more  commonly  and  perhaps  better,  one  plus  oti^, 
eq*uiU  two. 


To  BE  able  to  add  two,  three  or  four  columns 
of  figures  at  once,  is  deemed  by  many  to  be  a 
Herculean  task,  and  only  to  be  accomplished  by 
the  gifted  few,  or,  in  other  words,  by  mathemati- 
cal prodigies.  If  we  can  succeed  in  dispelling 
this  illusion,  it  will  more  than  repay  us ;  and  we 
feel  very  confident  that  we  can,  if  the  student 
will  lay  aside  all  prejudice,  bearing  steadily  in 
mind  that  to  become  proficient  in  any  new  branch 
or  principle  a  little  wholesome  application  is 
necessary.  On  the  contrary,  we  can  not  teach  a 
student  who  takes  no  interest  in  the  matter,  one 
who  will  always  be  a  drone  in  society.  Such 
men  have  no  need  of  this  principle. 

If  two,  three,  or  more,  columns  can  be  carried 
up  at  a  time,  there  must  be  some  law  or  rule  by 
which  it  is  done.  We  have  two  principles  of  Addi- 
tion; one  for  adding  short  columns,  and  one  for 
adding  very  long  columns.  They  are  much  alike, 
differing  only  in  detail.  When  one  is  thoroughly 
learned,  it  is  very  easy  to  learn  the  second. 
12 


ADDITION.  13 

ADDITION   TABLE. 

The  design  of  the  table  on  the  following  page 
is  to  familiarize  the  student  with  the  combination 
or  grouping  of  figui^es  so  as  to  enable  him  in- 
stantly to  see  or  read  the  result  without  stopping 
to  add  each  figure  separately. 

In  learning  this  table  avoid  spelling  the  figures, 
as  4  and  5  are  9,  but  take  in  the  result  9  as  soon 
as  the  eye  catches  the  combination — do  not  con- 
sider the  figures  4+5,  but  see  them  as  9.  To  il- 
lustrate: add  4+ 5 -f  6+2,  instead  of  saying  4  and 
5  are  9  and  6  are  15  and  2  are  17,  consider  the 
combinations  4+  5  as  9, 6+ 2  as  8 ;  thus  you  really 
have  but  9+8  to  add  instead  of  4+5+6  +  2,  pro- 
ducing a  saving  in  time  and  mental  work. 

The  science  of 

RAPID  OR  "LIGHTNING"  ADDITION 

Lies  in  the  ability  of  the  calculator  to  instantly 
see  or  take  in  the  result  of  two  or  more  figures 
regardless  of  their  combination,  without  stopping 
to  add  each  figure  separately,  i.  e.,  To  read  the 
result  of  figures  as  in  reading  a  book,  the  pro- 
nunciation of  a  word  is  known,  or  the  meaning 
of  a  sentence  without  the  necessity  of  spelling 
or  pronunciation  of  syllables. 

After  mastering  this  table  the  learner  will  be 
surprised  at  the  rapidity  he  can  add  a  column 
of  figures,  and  he  will  soon  find  himself  grouping 
or  combining  with  ease  and  accuracy  four  and 
five  figures  at  a  time,  instead  of  two  as  illus- 
trated by  the  table. 
2 


14   ORTON  &  Sadler's  calculator. 

TABLE    OF    ADDITION, 

Showing  the  combination  of  the  9  significant  figures^  in 

groups  of  two  only^  and  producing,  tohen  added 

together  J  results  from  \  to  IS. 


PrcKluced  by  coin- 
Pro-  bination  or  addi-  Pro- 
ducts,            tion  of  the  9             ducts, 
significant  figures. 


1 

= 

1 

~ 

I 

2 

= 

1 

1 

== 

2 

3 

== 

2 

1 

=r 

3 

4 

= 

3  2 
1  2 

= 

4 

5 

= 

43 
1  2 

= 

5 

6 

= 

5  4  3 
1  2  3 

= 

6 

r>  5  4 

1  2  3 
7  6  6  4 
1  2  3J^ 

rr'G  6 

12  3  4 


7  = 

8  = 

9  = 

in  «  9  8  7  G  5  ,^ 

*"  1  2  3  4  fS  '" 


11                      9  8  7  6  ,, 

"  =          2  3  4  5  =         11 

io  _  9  8  7  6  ,o 

*^                     3  4  5  6  ~         *^ 

13  =-         ??I  =13 


14 

« 

9  8  7 
5  67 

= 

14 

15 

=- 

9  8 
6  7 

= 

15 

16 

=- 

9  8 
7  8 

« 

16 

17 

= 

9 
8 

=3 

17 

18 

= 

9 
0 

= 

18 

ADDTTTON. 


N  B.  Tlie  above  process  of  addition  is  only  re* 
commended  for  beginners. 

Process. — For  adding  the  above  example,  cora- 
mence  at  the  bottom  of  the  right-hand  column. 
Add  thus  :  12,  16,  22 ;  then  carry  the  2  tens  to 
the  second  column,  then  add  thus,  8,  10,  18,  22, 
carry  the  two  hundreds  to  the  third  column,  and 
add  the  same  way,  9,  13,  16,  23.  Never  permit 
yourself  for  once  to  add  up  a  column  in  this  man- 
ner, 3  and  9  are  12  and  4  are  16,  and  6  are  22  ;  it 
is  just  as  easy  to  name  the  sura  at  once^  without 
naming  the  figures  you  add,  and  three  times  as 
rapid. 

15 


16   ORTON  &  Sadler's  calculator. 

ADDITION   OP  SHORT  COLUMNS  OF  FIGURES. 

Addition  is  the  basis  of  all  numerical  opera* 

tions,  and  is  used  in  all  departments  of  business, 

To  aid  the  business  man  in  acquiring  facility  and 

accuracy  in  adding  short  columns  of  figures,  the 

following  method  is  presented  as  the  best : 

Process. — Commence  at  the  bottom  of 
274 
040     the  right-hand  column,  add  thus:   16^  22, 

134     32 ;  then  carry  the  3   tens  to   the  second 

342     column;  then  add  thus:  7,  14,  25;  carry 

"^27     the  2  hundreds  to  the  third  column,  and 

^^^     add   the   same  way:  12,  16,  21.     In  this 

2152     '^ay  you  name  the  sum  of  two  figures  at 

once,  which  is  quite  as  easy  as  it  is  to  add  one 

figure  at  a  time.     Never  permit  yourself  for  once 

to  add  up  a  column  in  this  manner :  9  and  7  are 

16,  and  2  are  18   and  4  are  22,  and  6  are  28,  and 

4  are  32.     It  is  just  as  easy  to  name  the  result 

of  two  figures  at  once  and  four  times  as  rapid. 

The  following  method  is  recommended  for  the 

addition  of  long  columns  op  figures. 
In  the  addition  of  long  columns  of  figures 
which  frequently  occur  in  books  of  accounts,  in 
order  to  add  them  with  certainty,  and,  at  the 
name  time,  with  ease  and  expedition,  study  well 
the  following  method,  which  practice  will  rendei 
familiar,  easy,  rapid,  and  certain. 


ADDITION.  17 

THS    lAST    WAT    TO    ADD. 

EXAMPLE  2— EXPLANATION. 

Commence  at  9  to  add,  and  add  as  near  20  as  pos 
aible,  thus:  9+2+4+3:^18,  place  the  8  to  the 
right  of  the  3,  as  in  example ;  commence  at  6  to  7' 
add  6+4+8=18  ;  place  the  8  to  the  right  of  4 
the  8,  as  in  example ;  commence  at  6  to  add  6 
6+4+7=17  ;  place  the  7  to  the  right  of  the  3* 
7.  as  in  example ;  commence  at  4  to  add  4+  9 
9+3—16  ;  place  the  6  to  the  right  of  the  3,  4 
as  in  example ;  commence  at  6  to  add  6+4  7^ 
+7=17 ;  place  the  7  to  the  right  of  the  7,  4 
as  in  example;  now,  having  arrived  at  the  6 
top  of  the  column,  we  add  the  figures  in  the  8* 
new  column,  thus:  7+6+7+8+8=36  ;  place  4 
the  right  hand  figure  of  36,  which  is  a  6,  6 
under  the  original  column,  as  in  example,  and  3^ 
add  the  left  hand  figure,  which  is  a  3,  to  the  4 
number  of  figures  in  the  new  column;  there  2 
are  5  figures  in  the  new  column,  therefore  9 
3+5=8 ;  prefix  the  8  with  the  6,  under  the  — 
original  column,  as  in  example  ;  this  makes  86 
86,  which  is  the  sum  of  the  column. 

Remark  1. — If,  upon  arriving  at  the  top  of  the 
column,  there  should  be  one,  two  or  three  figure? 
whose  sum  will  not  equal  10,  add  them  on  to  the 
•urn  of  the  figures  of  the  new  column,  never  placing 


18        OKTON  &  Sadler's  calculator. 

an  extra  figure  in  the  new  column,  unless  it  be  aL 
excess  of  units  over  ten. 

Remark  2. — By  this  system  of  addition  you  can 
stop  any  place  in  the  column,  where  the  sum  of  the 
figures  will  equal  10  or  the  excess  of  10  ;  but  the 
addition  will  be  more  rapid  by  your  adding  as 
near  20  as  possible,  because  you  will  save  the  form- 
ing of  extra  figures  in  your  new  column. 
EXAMPLE— EXPLANATION. 

2+6+7=15,  drop  10,  place  the  5  to  the  right 
of  the  7;  6+5+4=15,  drop  10,  place  the  5  to 
the  right  of  the  4,  as  in  example ;  8+3+7=18, 
drop  10,  place  the  8  to  the  right  of  the  7,  4 
as  in  example;  now  we  have  an  extra  figure,  7* 
which  is  4  ;  add  this  4  to  the  top  figure  of  the  3 
new  column,  and  this  sum  on  the  balance  of  8 
the  figures  in  the  new  column,  thus:  4+8+  4* 
6+5=1:22 ;  place  the  right  hand  figure  of  22  5 
under  the  original  column,  as  in  example,  and  6 
add  the  left  hand  figure  of  22  to  the  num-  7' 
ber  of  figures  in  the  new  column,  which  are  6 
three,  thus  :  2+3=5 ;  prefix  this  5  to  the  2 
fip:ure  2,  under  the  original  colum:i ;  this  — 
uxakes  52,  which  is  the  sum  of  the  column.         52 


ADDITION.  19 

Rule.'— Far  adding  two  or  more  columris,  com- 
rn^mce  at  the  right  handj  or  units^  column;  proceed 
in  the  same  manner  as  in  adding  one  column ;  after 
the  sum  of  the  first  column  is  obtained^  add  all 
except  the  right  hand  figure  of  this  sum  to  the  second 
column,  adding  the  second  column  the  same  way  you, 
added  the  first;  proceed  in  like  manner  with  all  the 
columns,  always  adding  to  each  successive  column 
the  sum  of  the  column  in  the  next  lower  order,  minus 
the  right  hand  figure, 

N.  B.  The  small  figures  which  '^e  place  to  the 
right  of  the  column  when  adding  are  called  integers. 

The  addition  by  integers  or  by  forming  a  new 
column,  as  explained  in  the  preceding  examplea 
should  be  used  only  in  adding  very  long  columns 
of  figuies,  say  a  long  ledger  column,  where  the  foot- 
ings of  each  column  would  be  two  or  three  hundred, 
In  which  case  it  is  superior  and  much  more  easy 
than  any  other  mode  of  addition ;  but  in  adding 
short  columns  it  would  be  useless  to  form  an  extra 
column,  where  there  is  only,  say,  six  or  eight  fig- 
ures to  be  added.  In  making  short  additions,  the 
following  suggestions  will,  we  trust,  be  of  use  to 
the  accountant  who  seeks  for  information  on  this 
subject. 

In  the  addition  of  several  columns  of  figures, 
where  they  are  only  four  or  five  deep,  or  when 
their  respective  sums  will  range  from  twenty  ^ve 


20        ORTON  &  Sadler's  calculator. 

to  forty,  the  accountant  should  commence  with  the 
unit  column,  adding  the  sum  of  the  first  two  figures 
to  the  sum  of  the  next  two,  and  so  on,  naming  onlj 
the  results,  that  is,  the  sum  of  every  two  figures. 

In  the  present  example  in  adding  the  unit  346 
column  instead  of  saying  8  and  4  are  12  and  235 
5  are  11  and  '6  are  23,  it  is  better  to  let  the  T24 
eye  glide  up  the  column  reading  only,  8,  12,  598 
lY,  23;  and  still  better,  instead  of  making  a 
separate  addition  for  each  figure,  group  the  figures 
thus:  12  and  11  are  23,  and  proceed  in  like  man- 
ner with  each  column.  For  short  columns  this  is 
a  very  expeditious  way,  and  indeed  to  be  preferred ; 
but  for  long  columns,  the  addition  by  integers  is 
the  most  useful,  as  the  mind  is  relieved  at  intervals 
and  the  mental  labor  of  retaining  the  whole  amount, 
as  you  add,  is  avoided,  which  is  very  important  to 
any  person  whose  mind  is  constantly  employed  in 
various  commercial  calculations. 

In  adding  a  long  column,  where  the  figures  are 
of  a  medium  size,  that  is,  as  many  8s  and  9s  as 
there  are  2s  and  3s,  it  is  better  to  add  about  three 
figures  at  a  time,  because  the  eye  will  distinctly  see 
that  many  at  once,  and  the  ingenious  student  will 
in  a  short  time,  if  he  adds  by  integers,  be  able  to 
read  the  amount  of  three  figures  at  a  glance,  or  as 
quick,  we  might  say,  as  he  would  read  a  single 
figure. 


ADDITION.  21 

Here  we  begin  to  add  at  the  bottom  of  the     *26* 
anit  column  and  add  successively  three  fig-       ^^ 
ores  at  a  time,  and  place  their  respective       ^o^ 
sums,  minus  10,  to  the  right  of  the  last  fig-     954 
are  added ;  if  the  three  figures  do  not  make       62 
10,  add  on  more  figures;  if  the  three  figures       87* 
make  20  or  more,  only  add  two  of  the  fig-       ^f 
ures.     The  little  figures  that  are  placed  to       j^^ 4 
the  right  and  left  of  the  column  are  called     877 
integers.     The  integers  in  the  present  ex-       33 
ample,  belonging  to   the  units  column,  are       84'* 
4,  4,  5,  4,  6,  which  we  add  together,  making       ^^ 

23;  place  down  3  and  add  2  to  the  number  

of  integers,  which  gives  7,  which  we  add  to     803 
the  tens  and  proceed  as  before. 

Eeason. — In  the  above  example,  every  time  we 
placed  down  an  integer  w^e  discarded  a  ten,  and 
when  we  set  down  the  3  in  the  answer  we  dis- 
carded two  tens;  hence,  we  add  2  on  to  the  num- 
ber of  integers  to  ascertain  how  many  tens  were 
discarded;  there  being  5  integers  it  made  7  tens, 
which  we  now  add  to  the  column  of  tens;  on  the 
same  principle  we  might  add  between  20  and  30, 
always  setting  down  a  figure  before  we  got  to  30; 
then  every  integer  set  down  would  count  for  2  tens, 
being  discarded  in  the  same  way,  it  does  in  the 
present  instance  for  one  ten.  When  we  add  be- 
tween 10  and  20,  and  in  very  long  columns^  U 


22    ORTON  &  Sadler's  calculator. 

would  be  much  better  to  go  as  near  30  as  possible, 
and  count  2  tens  for  every  integer  set  down,  in 
which  case  we  would  set  down  about  one -half  as 
many  integers  as  when  we  write  an  integer  foi 
ev^ery  ten  we  discard. 

When  adding  long  columns  in  a  ledger  or  day- 
book, and  where  the  accountant  wishes  to  avoid  the 
writing  of  extra  figures  in  the  book,  he  can  place  a 
strip  of  paper  alongside  of  the  column  he  wishes 
to  add,  and  write  the  integers  on  the  paper,  and  in 
this  way  the  column  can  be  added  as  convenient 
almost  as  if  the  integers  were  written  in  the  book. 

Perhaps,  too,  this  would  be  as  proper  a  time  as 
any  other  to  urge  the  importance  of  another  good 
habit;  I  mean  that  of  making  'plain  figures.  Some 
persons  accustom  themselves  to  making  mere 
scrawls,  and  important  blunders  are  often  the  result. 
If  letters  be  badly  made  you  may  judge  from 
Buch  as  are  known;  but  if  one  figure  be  illegible, 
its  value  can  not  be  inferred  from  the  others.  The 
vexation  of  the  man  who  wrote  for  2  or  3  monkeys, 
and  had  203  sent  him,  was  of  far  less  importance 
than  errors  and  disappointments  sometimes  result- 
ing from  this  inexcusable  practice. 

We  will  now  proceed  to  give  some  methods  of 
proof.  Many  persons  are  fond  of  proving  the  cor- 
rectness of  work,  and  pupils  are  often  instructed 
to  do  so,  for  the  double  purpose  cf  giving  them 


ADDITION.  23 

exercise  in  nalculation  and  saving  their  teacher  the 
trouble  of  reviewing  their  work. 

There  are  special  modes  of  proof  of  elementary 
operations,  as  by  casting  out  threes  or  nines,  or  by 
changing  the  order  of  the  operation,  as  in  add- 
ing upward  and  then  downward.  In  Addition, 
some  prefer  reviewing  the  work  by  performing  the 
Addition  downward,  rather  than  repeating  the 
ordinary  operation.  This  is  better,  for  if  a  mis- 
take be  inadvertently  made  in  any  calculation, 
and  the  same  routine  be  again  followed,  we  are  very 
liable  to  fall  again  into  the  same  error.  If,  for 
instance,  in  running  up  a  column  of  Addition  you 
should  say  84  and  8  are  93,  you  would  be  liable,  rn 
going  over  the  same  again,  in  the  same  way  to 
slide  insensibly  into  a  similar  error ;  but  by  begin- 
ning at  a  different  point  this  is  avoided. 

This  fact  is  one  of  the  strongest  objections  to 
the  plan  of  cutting  off  the  upper  line  and  adding 
it  to  the  sum  of  the  rest,  and  hence  some  cut  off 
the  lower  line  by  which  the  spell  is  broken.  The 
most  thoughtless  can  not  fail  to  see  that  adding  a 
line  to  the  sum  of  the  rest,  is  the  same  as  adding  it 
in  with  the  rest. 

The  mode  off  proof  by  casting  out  the  nines 
and  threes  will  be  fully  explained  in  a  following 
ahapter. 

A  very  excellent  mode  of  avoiding  error  in  ad4- 


24   ORTON  &  Sadler's  calculator. 

ing  long  columDS  is  to  set  down  the  result  of  each 
column  on  some  waste  spot,  observing  to  place  the 
numbers  successively  a  place  further  to  the  left 
each  time,  as  in  putting  down  the  product  figures 
m  multiplication;  and  afterward  add  up  the 
amount.  In  this  way  if  the  operator  lose  his 
count  he  is  not  compelled  to  go  back  to  units,  but 
ofily  to  the  foot  of  the  column  on  which  he  is  op- 
erating. It  is  also  true  that  the  brisk  accountant, 
who  thinks  on  what  he  is  doing,  is  less  liable  to 
err  than  the  dilatory  one  who  allows  his  mind  to 
wander.  Practice  too  will  enable  a  person  to  read 
amounts  without  naming  each  figure,  thus  instead 
of  saying  8  and  6  are  14,  and  7  are  21  and  5  are  26. 
it  is  better  to  let  the  eye  glide  up  the  column,  read- 
ing only  8,  14,  21,  26,  etc.;  and,  still  further,  it  is 
quite  practicable  to  accustom  one's  self  to  group  87 
the  figures  in  adding,  and  thus  proceed  very  rap-  23 
idly.  Thus  in  adding  the  units'  column,  instead  45 
cf  adding  a  figure  at  a  time,  we  see  at  a  glance  62 
that  4  and  2  are  6,  and  that  5  and  3  are  8,  then  21 
6  and  8  arc  14 ;  we  may  then,  if  expert,  add  — 
constantly  the  sum  of  two  or  three  figures  at  a  time, 
and  with  practice  this  will  be  found  highly  advan- 
tageous in  long  columns  of  figures;  or  two  or  three 
columns  may  be  added  at  a  time,  as  the  practiced 
eye  will  see  that  24  and  62  are  86  almost  as  readily 
AS  that  4  and  2  are  6. 


ADDITION.  26 

Teachers  will  find  the  following  mode  of  match 
'  ng  lines  for  beginners  very  convenient,  as  they  can 
inspeoc  them  at  a  glance : 

Add  7654384 
8786286 
3408698 
2345615 
1213713 


23408696 


In  placing  the  above  the  lines  are  matched  in 
pairs,  the  digits  constantly  making  9.  In  the 
above,  the  first  and  fourth,  second  and  fifth  are 
matched;  and  the  middle  is  the  hey  line^  the  result 
being  just  like  it,  except  the  units'  place,  which  ia 
as  many  less  than  the  units  in  the  key  line  as  there 
are  pairs  of  lines;  and  a  similar  number  will  oc- 
cupy the  extreme  left.  Though  sometimes  used  as 
a  puzzle,  it  is  chiefly  useful  in  teaching  learners ; 
and  as  the  location  of  the  key  line  may  be  changed 
in  each  successive  example,  if  necessary,  the  arti- 
fice  could  not  be  detected.  The  number  of  lines 
U  necessarily  odd. 


a 


SHORT  METHODS  OF  MULTIPLICATION. 


Rule. — Set  down  the  smaller  factor  under  the 
larger,  units  under  units,  tens  under  tens.  Begin 
with  the  unit  figure  of  the  multiplier,  multiply  by  it, 
first  the  units  of  the  multiplicand,  setting  the  units 
of  the  product,  and  reserviny  the  tens  to  be  added 
to  the  next  product ;  now  multiply  the  tens  of  the 
multiplicand  by  the  unit  figure  of  the  multiplier^ 
and  the  unite  of  the  multiplicand  by  tens  figure  of 
26 


MULTIPLICATION.  27 

the  multiplier;  add  these  two  products  together ^  set- 
ting down  the  units  of  their  sum,  and  reserving  thi 
tens  to  he  added  to  the  next  product ;  now  multijoly 
the  tens  of  the  multiplicand  hy  the  tens  figure  of  tJu 
multiplier,  and  set  down  the  whole  amount.  This 
will  be  the  complete  product. 

Remark. — Always  add  in  the  tens  that  are  re- 
served as  soon  as  you  form  the  first  product. 

EXAMPLE  1.— EXPLANATION. 

1.  Multiply  the  units  of  the  multiplicand       24 
by  the  unit  figure  of  the  multiplier,  thus :       31 

1X4  is  4  ;  set  the  4  down  as  in  example.     

2.  Multiply  the  tens  in  the  multiplicand  by  744 
the  unit  figure  in  the  multiplier,  and  the  units  in 
the  multiplicand  by  the  tens  figure  in  the  multi- 
plier, thus  :  1x2  is  2;  3x4  are  12,  add  these  two 
products  together,  2-f-12  are  14,  set  the  4  down 
as  in  example,  and  reserve  the  1  to  be  added  to  the 
next  product.  3.  Multiply  the  tens  in  the  multi- 
plicand by  the  tens  figure  in  the  multiplier,  and 
add  in  the  tens  that  were  reserved,  thus:  3x2  are 
6,  and  6+1=7 ;  now  set  down  the  whole 
amount,  which  is  7. 

EXAMPLE  2.— EXPLANATION. 

1.  Multiply  units  by  units,   thus:  4x3         53 
are  12,  set  down  the  2  and  reserve  the  1  to         84 

carry.    2.  Multiply  tens  by  units,  and  units 

by  tens,  and  add  in  the  one  to  carry  on  the     4452 


28        ORTON  &  Sadler's  calculator. 

arst  product,  then  add  these  two  products  together, 
thus:  4X5  are  20+1  are  21,  and  8x3  are  24, 
and  21-f-24  are  45,  set  down  the  5  and  reserve 
the  4  to  carry  to  the  next  product.  3.  Multiply 
tens  by  tens,  and  add  in  what  was  reserved  to  carry, 
thus:  8x5  are  40-[-4  are  44,  now  set  down  the 
whole  amount,  which  is  44. 

EXAMPLE  3.— EXPLANATION. 

5x3  are  15,  set  down  the  5  and  carry  the  43 
1    to  the  next   product;   5X4  are   20=1         25 

are  21;  2x3  are  6,  21+6  are  27,  set  down     

the  7  and  carry  the  2;  2x4  are  8+2  are  1075 
10  ;  now  set  down  the  whole  amount. 

When  the  multiplicand  is  composed  of  three  fig- 
ures, and  there  are  only  two  figures  in  the  multi- 
plier, we  obtain  the  product  by  the  following 

Rule, — Set  down  the  smaller  factor  under  the 
larger  J  units  under  units,  tens  under  tens ;  now  muU 
tiply  the  first  upper  figure  hij  the  unit  figure  of  the 
multiplier^  setting  down  the  units  of  the  product,  and 
reserving  the  tens  to  he  added  to  the  next  product; 
now  multiply  the  second  upper  hy  units,  and  the  fij-si 
upper  hy  tens,  add  these  two  products  together,  set- 
ting down  the  units  figure  of  their  sum,  and  reserv- 
ing the  tens  to  carry,  as  hefore ;  now  multiply  the 
third  upper  hy  units,  and  the  second  upper  hy  tens^ 
add  these  two  products  together,  setting  down  the 
units  figure  of  their  sum,  and  reserving  the  tens  to 


MULTIPLICATION.  20 

carry,  as  usual ;  now  multiply  the  third  upper  hy 
tens,  add  in  the  reserved  figure,  if  there  is  one,  and 
set  down  the  whole  amount.  This  will  he  (he  com  - 
pleie  product. 

Remark. — One  of  the  principal  errors  with  the 
beginner,  in  this  system  of  multiplication,  is 
neglecting  to  add  in  the  reserved  figure.  The  stu- 
dent must  bear  in  mind  that  the  reserved  figure  l<i 
added  on  to  the  first  product  obtained  after  the  set- 
ting down  of  a  figure  in  the  complete  product. 

EXAMPLE  1.— EXPLANATION. 

M\iltiplj  first  upper  by  units,  5x3  are  123 
15,  set  down  the  5,  reserve  the  1  to  carry         45 

to  the  next  product;  now  multiply  second     

upper  by  units  and  first  upper  by  tens,  5X2  5535 
are  10-f  1  are  11,  4X3  are  12,  add  these 
products  together;  11-|-12  are  23,  set  down  the  3, 
reserve  the  2  to  carry ;  now  multiply  third  upper 
by  units,  and  second  upper  by  tens,  add  these  two 
products  together,  always  adding  on  the  reserved 
figure  to  the  first  product;  5x1  are  5-|-2  are  7, 
4X2  are  8,  and  7-f-8  are  15,  set  down  the  5,  re- 
serve the  1 ;  now  multiply  third  upper  by  tens, 
and  set  down  the  whole  amount ;  4X 1  are  4-|-l  are 
5,  set  down  the  5.  This  will  give  the  comple>« 
product. 


30        ORTON  &  Sadler's  calculator. 

Multiply  32  by  45  in  a  single  line. 

Hero  we  multiply  5x2  and  set  down  and  32 
carry  as  usual ;  then  to  what  you  carry  add         45 

5X3  and  4X  2,  which  gives  24;  set  down     

4  and  carry  2  to  4x3,  which  gives  14  and     1440 
completes  the  product. 

Multiply  123  by  456  in  a  single  line. 

Here  the  first  and  second  places  are  123 
found  as  before;  for  the  third,  add  6X1,  456 

5X2,  4X3,  with  the  2  you  had  to  carry,  

making  30  ;  set  down  0  and  carry  3 ;  then  56088 
drop  the  units'  place  and  multiply  the 
hundreds  and  tens  crosswise,  as  you  did  the  tens 
and  units,  and  you  find  the  thousand  figure ;  then, 
dropping  both  units  and  tens,  multiply  the  4X1, 
adding  the  1  you  carried,  and  you  have  5,  which 
completes  the  product.  The  same  principle  may 
DC  extended  to  any  number  of  places ;  but  let  each 
Btep  be  made  perfectly  familiar  before  advancing 
to  another.  Begin  with  two  places,  then  take  three, 
then  four,  but  always  practicing  some  time  on  each 
number,  for  any  hesitation  as  you  progress  will, 
confuse  you. 

N.  B.  The  following  mode  of  multiplying  num- 
bers will  only  apply  where  the  sum  of  the  two  la&l 
or  unit  figures  equal  ten,  and  tho  other  figures  in 
both  factors  are  the  same. 


MULTIPLICATION.  3^1 

CONTKACTIONS  IN  MULTIPLICATION. 

To  multiply  when  the  unit  figures  added  equal 
*'I0)  and  the  tens  are  alike  as  72  ^^18,  dc, 

1st.  Multiply  the  units  and  set  down  the  result. 

2d.  Add  1  to  either  number  in  tens  place  and 
multiply  by  the  other,  and  you  have  the  complete 
product. 

EXAMPLE  PROCESS. 

Here  because  the  sum  of  the  units  4  and  6  86 
are  ten  and  the  tens  are  alike  ;  we  simply  say       84 

4  times  6  are  24,  and  set  down  both  figures  of    

the  product ;  then  because  4  and  6  make  ten  we  '^224 
add  1  to  8,  making  9,  and  9  times  8  are  72,  which 
completes  the  product. 

Note. — If  the  product  of  units  do  not  contain  ten  tbe  plac« 
of  tens  must  be  filled  with  a  cipher 

The  above  rule  is  useful  in  examples  like  the  fd- 
lowing  : 

2.  What  will  93  acres  of  land  cost  at  97  dollars 
per  acre?  Ans.  $9021. 

3.  What  will  89  pounds  of  tea  cost  at  81  cents 
per  pound?  Ans.  $72.09. 

In  the  above  the  product  of  9  hi/ I  did  not  amourU 
to  ten^  therefore  0  is  placed  in  tens  place. 

4.  Multiply  998  by  902.  Ans.  990016. 
In  the  above,  because  2  and  8  are  10,  ive  add  1  to 

99,  making  100;  then  100  times  99  are  9900 
3 


32  ORTON    &   SADLER'S    CALCULATOR^ 

EXAMPLE  EIGHTEENTH. 
Multiply  79  by  71  in  a  single  line. 
Here  we  multiply  IX^*  and  set  down  the       79 
result,  then  we   multiply  the  7  in  the  mul-       71 

tiplicand,  increased    by  1   by  the  7  in   the  

multiplier,  7X8,  which  gives  56  and  t>om-  6609 
pletes  the  product. 

EXAMPLE. 

Multiply  197  by  193  in  a  single  line. 

Here  we  multiply  3x7  and  set  down  the  197 
result,    then    we    multiply  the    19  in  the       193 

multiplicand,  increased  by  1   by  the  19  in 

the  multiplier,  19x20,  which  gives  380  38021 
4nd  completes  the  product. 

EXAMPLE. 

Multiply  996  by  994  in  a  single  line. 

Here  we  multiply  4x6  and  set  down  996 
the    result,    then  we  multiply  the  99    in         994 

the  multiplicand,  increased  by  1  by  the  

99  in  the  multiplier,  99x100,  which  990024 
gives  9900  and  completes  the  product. 

EXAMPLE. 

Multiply  1208  by  1202  in  a  single  line. 

Here  we  multiply  2x8  and  set  down  1208 
the  result,  then  we  multiply  the  120  in         1202 

the  multiplicand,  increased  by  1  by  tbc 

120  in  the  multiplier,  120x121,  which  1452016 
gives  14520  and  completes  the  product. 


^      MULTIPLICATION.  33 

CDEIOUS  AND  USEFUL  CONTRACTIONS. 

To  multiply  any  number,  of  two  figures,  by  11, 
Hole.--  Write  the  sum  of  the  figures  between  them 

1.  xVlultipiy  45  by  11  Ans.  495. 
Here  4  and  5  are  9,  which  write  between  4  &  5 

2.  Multiply  34  by  11.  Ans.  61A 
N.  B.   When  the  sum  of  the  two  figures  is  over 

9,  incretise  the  left-hand  figure  by  the  1  to  carry. 

3.  Multiply  87  by  11.  Ans.  957 

To  square  any  number  of  9s  instantaneously. 
and  without  multiplying. 

Rule. —  Write  down  as  many  98  less  one  as  there 
are  9s  in  the  given  number ^  an  8,  as  many  0«  as 
9«,  and  a  1. 

4.  What  is  the  square  of  9999  ?    Ans.  99980001. 
Explanation. — We  have  four  9s  in  the  given 

number,  so  we  write  down  three  9s,  then  an  8, 
then  three  Os,  and  a  1. 

5.  Square  999999.  Ans.  999998000001. 

To  square  any  number  ending  in  5, 

Rule. — Omit  the  5  and  multiply  the  number,  as 

it  will  then  stand  by  the  next  higher  number^  and 

annex  25  to  the  product, 

6.  What  is  the  square  of  75  ?  Ans.  5625. 
Explanation. — We  simply  say,  7  times  8  are 

56,  to  which  we  annex  25. 

7.  What  is  the  square  of  95?  Ans.  9026 


TABLE   OF   SQUARES, 

PBOM  1  TO  104. 


l«-r  1 

272=  729 

532=2809 

'i92=  6241 

22=     4 

282=  7g4 

542=2916 

802=  6400 

32=     9 

292=  841 

552=3025 

812=  6561 

42=  16 

302=  900 

562=3136 

822=  6724 

52=  25 

312=  9gi 

572=3249 

832=  6889 

62=  36 

322=1024 

582=3364 

842=  7056 

72=  49 

332=1089 

592=3481 

852=  7225 

82=  64 

342=1156 

602=3600 

862=  7396 

92=  81 

352=1225 

612=3721 

872=  7569 

102=100 

362=1296 

622=3844 

882=  7744 

112=121 

372=1369 

632=3969 

892=  7921 

122=144 

382=1444 

642=4096 

902=  8100 

132=169 

392=1521 

652^4225 

912=  828L 

142=196 

402=1600 

662=4356 

922=  8464 

152=225 

412=1681 

672=4489 

932=  8649 

162=256 

422=1764 

682=4624 

942=  8836 

172=289 

432=1869 

692=4761 

952=  9025 

182=324 

442=1936 

702--4900 

962=  9216 

192=361 

452=2025 

712=5041 

972=  9409 

202=400 

462=2116 

722=5184 

982=  9604 

212=441 

472=2209 

732=5329 

992=  9801 

222=484 

482=2304 

742=5476 

1002=10000 

232=529 

492=2401 

752=5625 

1012=10201 

242=576 

502=2500 

762=5776 

1022=10404 

252=625 

512=2601 

772=5929 

1032=10609 

262=676 

522=2704 

782=6084 

1042=10816 

Note. — To  become  familiar  with  the  numbers  shown  in 
the  above  table,  from  1  to  25,  requires  but  little  study  and 
application  upon  the  part  of  the  pupil,  and  will  prove  0/ 
great  benefit  in  mathematical  calculations. 

34 


FRACTIONS 

Are  one  or  more  of  the  equal  parts  into  which  a  unit  or 
whole  thing  is  divided. 

All  fractions  express  the  division  of  units  or  things. 

The  fractional  terms  are,  numerator  and  denominator. 

The  Numerator  expresses  the  number  of  parts  or  units 
taken  (it  is  therefore  the  dividend),  and  is  written  above 
the  line. 

The  Denominator  expresses  the  division  of  the  equal 
parts  or  units  (it  is  therefore  the  divisor),  and  is  written 
below  the  line. 

Fractions  are  written  and  read  as  follows : 

i  or  one-half,  J  or  one-third,  |  or  three- fourths. 

The  Quotient  produced  from  dividing  the  numerator  by 
the  denominator  of  a  fraction  is  its  value. 

Thus,  —  I  =-  3  —  L2  =:^  3  —  ^^  -=  9 

The  vahie  of  a  fraction  is 'less  than  1  when  the  numera- 
tor is  less  than  the  denominator,  and  equals  or  exceeds  1 
when  the  numerator  equals  or  is  greater  than  the  denomi- 
nator. 

GENERAL    PRINCIPLES    GOYERM\G    FRACTIONS. 

To  increase  or  multiply  a  fraction, 

Multiply  the  numerator  or  divide  the  denoviinater. 

To  decrease  or  divide  a  fraction, 

jylvlde  the  munerator  or  multiply  the  denoviinator. 

Multiplying  or  dividing  both  terms  of  a  fraction  by  the 
same  number  does  not  change  its  value. 

Fractions  may  be  reduced,  added,  subtracted,  multi- 
plied, and  divided. 

35 


S6        ORTON  <&  Sadler's  calculator. 

Mental  Operations  in  Fractions. 

To  square  any  number  containing  |,  as  6^,  9-^, 

Rule. — Multiply  the  whole  number  hy  the  next 
hiyher  whole  number,  and  annex  ^  to  the  product. 

Ex.  1.  What  is  the  square  of  7^?     Ans.  m^. 

We  simply  say,  7  times  8  are  56,  to  which  we 
addf 

2.  What  will  9 J  lbs.  beef  cost  at  9^  cts.  a  lb.? 

3.  What  will  12^  yds.  tape  cost  at  12^ cts.  a  yd.? 

4.  What  will  5^  ]bs.  nails  cost  at  5|  cts.  a  lb.  ? 

5.  What  will  11^  yds.  tape  cost  at  11^  cts.  a  yd.? 

6.  What  will  19^  bu.  bran  cost  at  19^  cts.  a  bu.7 
Reason. — We  multiply  the  whole  number  by 

the  next  higher  whole  number,  because  half  of  any 
number  taken  twice  and  added  to  its  square  is  the 
same  as  to  multiply  the  given  number  by  one  moie 
than  itself.  The  same  principle  will  multiply  any 
two  like  numbers  together,  when  the  sum  of  the 
fractions  is  one,  as  8^  by  8|,  or  11 J  by  11|,  etc 
It  is  obvious  that  to  multiply  any  number  by  any 
two  fractions  whose  sum  is  ONE,  that  the  sum  of  the 
products  must  be  the  originr''  number y  and  adding  the 
number  to  its  square  is  simply  to  multiply  it  by 
ONE  more  than  itself;  for  instance,  to  multiply  7^ 
by  7|,  we  simply  say,  7  times  8  are  5f>^  and  then, 
to  complete  the  multiplication,  we  add,  of  course, 
the  product  of  the  fractions  (J  times  J  arc  3^), 
making  5 6^5  the  answer. 


MULTIPLICATION   OP   FRACTIONS.  37 

WTiere  the  sum  of  the  Fractions  is  One. 

To  multiply  any  two  like  numbers  together  when 
the  sum  of  the  fractions  is  one. 

Rule. — Multiple/  the  whole  number  hy  the  next 
higher  whole  number;  after  whichy  add  the  product 
of  tlie  fractions, 

N.  B.  In  the  following  examples,  the  product  of 
the  fractions  are  obtained  yrrs^  for  convenience. 

PRACTICAL  EXAMPLES  FOR  BUSINESS  MEN. 

Multiply  3|  by  3^  in  a  single  line. 

Here  we  multiply  |X|,  which  gives  y^^,  3^ 

and  set  down  the  result;  then  we  multiply  3| 

the    3    in    the    multiplicand,   increased    by  

unity,  by  the  3  in  the  multiplier,  3x4,  12 ^^ 
which  gives  12  and  completes  the  product. 

Multiply  7f  by  7f  in  a  single  line. 

Here  we  multiply  |Xf)  which  gives  /^,  7| 

and  set  down  the  result;  then  we  multiply  7| 

the  7  in  the  multiplicand,  increased  by  unity,  

by  the  7  in  the  multiplier,  7x8,  which  gives  56^ 
56,  and  completes  the  product. 

Multiply  11^  by  11|  in  a  single  line. 

Here  we  multiply  f  X^,  which  gives  J,  and  11 J 

aet  down  the  result;  then  we  multiply  the  11  11 J 

in  the  multiplicand,  increased  by  unity,  by  

the  11  in  the  multiplier,  11x12.  which  gives  132 J 
132.  and  completes  the  product. 


38        ORTON  &  Sadler's  calculator. 

EXAMPLE. 

Multiply  16f  by  16J  in  a  single  line. 

Here  we  multiply  JXf  which  gives  J,  and  16| 
set  down  the  result,  then  we  multiply  the       16J 

16    m    the     multiplicand,    increased     by > 

unity  by  the  16  in  the  multiplier,  16x17,     2721 
which  gives  272  and  completes  the  product. 
EXAMPLE. 

Multiply  29J  by  29^  in  a  single  line. 

Here  we  multiply  JXi  which  gives  J,  29J 
and  set  down  the  result,  then  we  multiply       29^ 

the  29  in  the  multiplicand,  increased  by     

unity   by  the   29  in  the  multiplier,   29  x     870| 
30,  which  gives  870  and  completes  the  pro- 
duct. 

EXAMPLE. 

Multiply  999§  by  999f  in  a  single  line. 

Here  we  multiply  fXf?  which  gives  999| 

J|,  and  set  down   the  result,  then  we  999f 

multiply  the  999  in  the  multiplicand, 

increased  by  unity  by  the  999  in  the     999000JJ 
multiplier,    999x1000,    which    gives 
999000  and  completes  the  product. 

Note. — The  system  of  multiplication  introduced 
in  the  preceding  examples,  applies  to  all  numbers. 
Where  the  sum  of  the  fractions  is  owe,  and  the  whole 
numbers  are  alike,  or  differ  by  one,  the  learner  if 
requested  to  study  well  these  useful  properties  of 
numboTP 


MULTIPLICATION   OF   FRACTIONS.  39 

Where  the  sum  of  the  Fractioiis  is  One. 

To  multiply  any  two  numbers  whose  diflerenoe 
is  onSy  and  the  sum  of  the  fractions  is  one. 

Rule. — Multiply  the  larger  number,  increased  ht^ 
ONE,  bi/  the  smaller  number;  then  square  the  frac- 
tion of  the  larger  number,  and  subtrac*  its  square 
from  ONE. 

PRACTICAL  EXAMPLES  FOR  BUSINESS  MEN. 

1.  What  will  9^  lbs.  sugar  cost  at  8|  cts.  a  lb.  ? 
Here  we  multiply  9,  increased  by  1,  by  8,         91 

thus,  8x10  are  80,  and  set  down  the  result;         gl 

then  from  1  we  subtract  the  square  of  -J,     

thus,  \  squared  is  J^,  and  1  less  ^  is  f|.        ^^H 

2.  What  will  8|  bu.  coal  cost  at  7^  cts.  a  bu.? 
Here  we  multiply  8,  increased  by  1,  by  8| 

7j  thus,  7  times  9  are  63,  and  set  down  the  7  J 

result ;  then  from  1  we  subtract  the  square       ~"^ 
of  I,  thus,  I  squared  is  |,  and  1,  less  |,  is  |.  » 

3.  What  will  11-j^  bu.  seed  cost  at  $10}  J  a  bu.? 
Here  we  multiply  11,  increased  by  1,  by 

10,  thus,  10   times   12  are   120,  and  set     ^^A 
down  the  result;  then  from  1  we  subtract     ^^T? 
the  square  of  j^g,  thus,  ^  squared  is  j^^,  lOQlta 
and  1  less  ^-J^  is  f«-f. 

4.  How  many  square  inches  in  a  floor  99|  in 
wiie  and  98|  in.  long?  Ans.  9800^4 


40        ORTON  &  Sadler's  calculator. 

METHOD  OF  OPERATION. 
EXAMPLE. 

jdultiply  6J  by  6J  in  a  single  line. 

Here  we  add  6J+ J,  which  gives  6 J ;  this  6} 
multiplied  by  the  6  in  the  multiplier,  6J 
6X6^,  gives  39,  to  which  we  add  the  pro-  — 
duct  of  the  fractions,  thus  JX  J  gives  j^^,  added 
89j'g  to  39  completes  the  product. 

EXAMPLE. 

Multiply  11 J  by  llf  in  a  single  line. 

Here  we  would  add  llJ-[-f)  which  gives  11J 
12;  this  multiplied  by  the  11  in  the  multi-     Jl| 

plier  gives  132,  to  which  we  add  the  product  

of  the  fractions,  thus  f  Xi  gives  j\y  which  132 j^^ 
added  to  132  completes  the  product. 

EXAMPLE. 

Multiply  12J  by  12f  in  a  single  line. 

Here  we  add  12^-|-f,  which  gives  13J;  12J 
this  multiplied  by  the  12  in  the  multiplier,     12| 

12X13J,  gives  159,  to  which  add  the  pro-  

duct  of  the  fractions,  thus  fX^  gives  §,  159j 
which  added  to  159  completes  the  product. 


MULTIPLICATION   OF   FRACTIONS.  41 

Where  the  Fractions  have  a  Like  Denominator, 
To  multiply  any  two  like  numbers  together,  each 
3f  which  has  a  fraction  with  a  like  denominator,  as 
l|  by  4J,  or  11^  by  llf,  or  10|  by  lOJ,  eic. 

Rule. — Add  to  the  multiplicand  the  fraction  of 
the  multiplier,  and  multiple/  this  sum  by  the  whole 
number;  afterwhich,  add  the  product  of  the  fractions. 

PRACTICAL  EXAMPLES  FOR  BUSINESS  MEN. 

N.  B.  In  the  following  example,  the  sum  of  the  frao- 
tlons  is  ONE. 

1.  What  will  9f  lbs.  beef  cost  at  9^  cts.  a  lb.? 

The  sum  of  9|  and  \  is  ten,  so  we  simply     ^1 

say,  9  times   10  are    90;    then  we  add  the  f_ 

product  of  the  fractions,  \  times  |  are  j^^r.       ^O^g 
N.  B.  In  the  following  example,  the  sum  of  the  frac 
tlons  is  less  than  one. 

2    What  will  8^1^  yds.  tape  cost  at  8f  cts.  a  yd.  ? 

The  sum  of  8J  and  |  is  8|,  so  we  simply       ^ 

say,  8  times  8J  are  70;    then  we  add   the  _4 

product  of  the  fractions,  \  times  \  are  ^  or  ^.     70^ 

N.  B.  In  the  following  example,  the  sum  of  the  frac- 
tions is  greater  than  one. 

3.  What  will  4f  yds.  cloth  cost  at  $4|  a  yd.? 

The  sum  of  4|  and  |  is  5|^,  so  we  simply  4| 
say,  4  times  5J  are  21 ;  then  we  add  the  » 
product  of  the  fractions,  ^  times  |^  are  |^.       21 J^ 

N.  B.  "Where  the  fractions  have  different  ienominators. 
reduce  them  to  a  common  denominator. 


42        ORTON  &  Sadler's  calculator. 

Rapid  Process  of  Multiplying  Mixed  Kumhers, 

A  valuable  and  useful  rule  for  the  accountant  in 
the  practical  calculations  of  the  counting-room. 

To  multiply  any  two  numbers  together,  each  of 
which  involves  the  fraction  ^,  as  7^  by  9^,  etc., 

Rtjle. —  To  the  product  of  the  whole  numbers  add 
half  their  sum  plus  ^, 

EXAMPLES  FOR  MENTAL  OPERATIONS. 

1.  What  will  3^doz.  eggs  cost  at  7^  cts.  a  doz.? 
Here  the  sum  of  7  and  3  is  10,  and  half  this     31 

sum  is  5,  so  we  simply  say,  7  times  3  are  21     7| 

and  5  are  26,  to  which  we  add  4-.  

^  264 

N.  B.  If  the  sum  be  an  odd  number,  call  it  one  less 

to  make  it  even,  and  in  such  cases  the  fraction  must  be  |. 

2.  What  will  11^  lbs.  cheese  cost  at  9^  ets.  a  lb.? 

3.  What  will  8^  yds.  tape  cost  at  15^  cts.  a  yd.? 

4.  What  will  7^  lbs.  rice  cost  at  13|  cts.  a  lb.? 
6.  What  will  10^  bu.  coal  cost  at  12|  cts.  a  bu.? 
Reason. — In  explaining  the  above  rule,  we  add 

half  their  sum  because  half  of  either  number  added 
to  half  the  other  would  be  half  their  sum,  and  we 
add  ^  because  ^  by  ^  is  ^.  The  same  principle 
will  multiply  any  two  numbers  together,  each  of 
which  has  the  same  fraction;  for  instance,  if  the 
fraction  was  ^,  we  would  add  one-third  their  sum  ; 
if  |,  we  would  add  three-fourths  their  sum,  etc.; 
and  then,  to  complete  the  multiplication,  we  would 
add.  of  course,  the  product  of  the  fractions. 


MULTIPLICATION   OF   FRACTIONS.  43 

GENERAL  RULE 
For  multiplying  any  two  numbers  together,  each 

of  <7bich  involves  the  same  fraction. 

To  the  product  of  the  whole  numhers^  add  the 
product  of  their  sum  by  either  fraction ;  after  which^ 
add  thf'  product  of  their  fractions. 

EXAMPLES  FOR  MENTAL  OPERATIONS. 

1.  What  will  llf  lbs.  rice  cost  at  9f  cts.  a  lb.? 
Here  the  sum  of  9  and  11  is  20,  and  three-     i\s 

fourths  of  this  sum  is  15,  so  we  simply  say,       9} 

9  times  11  are  99  and  15  are  114,  to  which  

we  add  the  product  of  the  fractions  (3^).       ■'^■^^^rfl 

2.  What  will  7|  doz.  eggs  cost  at  8|  cts.  a  doz.  ? 

3.  What  will  6|  bu.  coal  cost  at  6|  cts.  a  bu.  ? 

4.  What  will  45|  bu.  seed  cost  at  3|  dol.  a  bu.? 
5    Af  hat  will  3|  yds.  cloth  cost  at  5f  dol.  a  yd.  ? 

6.  What  will  17|  ft.  boards  cost  at  13|  cts  a  ft.? 

7.  What  will  18|  lbs.  butter  cost  at  18|  cts.  a  lb.  ? 

N.  B.  If  the  produt*t  of  the  sum  by  either  frac- 
tion in  a  whole  number  with  a  fraction,  it  is  better 
to  reserve  the  fraction  until  we  are  through  with 
the  whole  numlers,  and  then  add  it  to  the  product 
of  the  fractions;  for  instance,  to  multiply  3:J-  by  7|, 
we  find  the  sum  of  7  and  3,  which  is  10,  and  one- 
fourth  of  this  sum  is  2^;  setting  the  |  down  in 
gome  waste  spot,  we  simply  say,  7  times  3  are  21 
and  2  are  23 ;  then,  adding  the  i  to  the  product  of 
the  fractions  (^^),  gives  -j^g^,  making  23^^,  Ans. 


44        ORTON  &  Sadler's  calculator. 

t 

Rapid  Process  of  Multiplying  all  Mixed  Numbers. 
N.  B.  Let  the  student  remember  that  this  is  a 
general  and  universal  rule. 

GENELAL  RULE. 

To  multiply  any  two  mixed  numbers  together, 

1st.  Multiply  the  whole  numbers  together. 

2d.    Multiply  the  upper  digit  by  the  lower  fractio^i. 

3d.    Multiply  the  lower  digit  by  the  upper  fraction. 

4th.  Multiply  the  fractions  together. 

5th.  Add  these  FOUR  products  together, 

N.  B.  This  rule  is  so  simple,  so  useful,  and  so  true  that 
svery  banker,  broker,  merchant,  and  clerk  should  post  i^ 
up  for  reference  and  use. 

PRACTICAL  EXAMPLES  FOR  BUSINESS  MEN 
N.  B.  The  following  method  is  recommended  to  begin- 
ners: 

Example.— -Multiply  12|  by  9|.  12f 

1st    We    multiply   the    whole   numbers.       ^f 
2d.    Multiply  12  by  f  and  write  it  down.  ;[08 
3d.    Multiply    9  by  |  and  write  it  down.       9 
4th.  Multiply    |  by  |  and  write  it  down.       6 
5th.  Add  these  four  products   together,  __A 
and  we  have  the  complete  result.       123j^ 
N.  B.  When  the  student  has  become  familiar 
T^ith  the  above  process,  it  is  better  to  do  the  intei- 
nediate  work  in  the  head,  and,  instead  of  setting 
iown  the  partial  products,  add  them  in  the  mind 
as  you  pass  along,  and  thus  proceed  very  rapidly. 


MULTIPLICATION   OF    MIXED    NUMBERS.      45 

Multiply  8|  by  lOf 

Here  we  simply  say  10  times  8  arc  80       8J 
and  I  of  8  is  2,  making  82,  and  |  of  10  is     10} 

2,  which  makes  84 ;  then  ^  times  I  is  2^^^,     

making  84^^^  the  answer.  ^'hh 

PRACTICAL  BUSINESS  METHOD 

For  Multiplying  all  Mixed  Numbers, 

Merchants,  grocers,  and  business  men  generally, 
in  multiplying  the  mixed  numbers  that  arise  Id 
the  practical  calculations  of  their  business,  only 
care  about  having  the  answer  correct  to  the  near- 
est cent ;  that  is,  they  disregard  the  fraction. 
When  it  is  a  half  cent  or  more,  they  call  it  an- 
other cent ;  if  less  than  half  a  cent,  they  drop  it. 
And  the  object  of  the  following  rule  is  to  show  the 
business  man  the  easiest  and  most  rapid  process  of 
finding  the  product  to  the  nearest  unit  of  any  two 
numbers,  one  or  both  of  which  involves  a  fraction. 
GENERAL  RULE. 

To  multiply  any  two  numbers  to  the  nearest  unit, 

1st.  Multiply  the  whole  number  in  the  multiplicand 
by  the  fraction  in  the  multiplier  to  the  nearest  unit, 

2d.  Multiply  the  whole  number  in  the  multiplier  hy 
the  fraction  in  the  multiplicand  to  the  nearest  unit 

3d.  Multiply  the  whole  numbers  together  and  add 
the  three  products  in  your  mind  as  you  proceed. 

N.  B.  In  actual  business  the  work  can  generally  be 
done  mentally  for  only  easy  fractions  occiw  in  business 


46        ORTON  &  Sadler's  calculator. 

N  B.  This  rule  is  so  simple  and  so  true,  according  tA 
all  business  usage,  that  every  accountant  should  make 
himsel'f  perfectly  familiar  with  its  application.  There 
being  no  such  thing  as  a  fractian  to  add  in,  there  is 
scarcely  any  liability  to  error  or  mistake.  By  no  other 
arithmetical  process  can  the  result  be  obtained  by  so  fer 
figures. 

EXAMi'LES  FOR  MENTAL  OPERATION. 
EXAMPLE    FIRST. 

Multiply  11^  by  8|  by  business  method.      11^ 
Here  :|  of  1 1  to  the  nearest  unit  is  3,  and  ^  of      8J 

8  to  the  nearest  unit  is  3,  making  6,  so  we  sim-     • 

ply  say,  8  times  11  are  88  and  6  are  94,  Ans.     94 

Reason. — \  of  11  is  nearer  3  than  2,  and  J  of  8  is  nearer 
3  than  2.     Make  the  nearest  whole  number  the  quotient, 

EXAMPLE    SECOND. 

Multiply  7|  by  9|  by  business  method. 

Here  |  of  7  to  the  nearest  unit  is  3,  and  J  7| 

of  9  to  the  nearest  unit  is  7 ;  then  3  plus  7  9| 

is  10,  so  we  simply  say,  9  times  7  are  63  and  

10  are  73,  Ans.  73 

EXAMPLE   THIRD. 

Multiply  23-^  by  19^  by  business  method. 

Here  ^  of  23  to  the  nearest  unit  is  6,  and  23| 
^  of  19  to  the  nearest  unit  is  6  ;  then  6  plus     19| 

6  is  12,  so  we  simply  say,  19  times  23  are 

437  and  12  are  449,  Ans.  "^^^ 

N.  B.  In  multiplying  the  whole  numbers  together,  sA 
ways  use  the  single-line  method. 


MULTIPLICATION   OF   MIXED   NUMBERS.      47 
EXAMPLE    FOUETH. 

Multiply  128|  by  25  by  business  method. 
Here  |  of  25  to  the  oearest  unit  is  17,  so     128 1 
we  simply  say,  25  times  128  are  3200  and       ^^ 

17  are  3217,  the  answer.  3217 

PRACTICAL  EXAMPLES  FOR  BUSINESS  MEN. 

1.  What  is  the  cost  of  17^  lbs.  sugar  at  18|  cts. 
per  lb.  ? 

Here  |  of  17  to  the  nearest  unit  is  13,       17| 
and  ^  of  18;  is  9  13  plus  9  is  22,  so  we       18| 

simply  say,  18  times  17  are  306  and  22  are 

328,  the  answer.  ^^'^^ 

2.  What  is  the  cost  of  11  lbs.  5  oz.  of  butter  a*. 
831  cts.  per  lb.? 

Here  ^  of  11  to  the  nearest  unit  is  4,       H^ 
and  ^  of  33  to  the  nearest  unit  is  10 ;       331 

then  4  plus  10  is  14,  so  we  simply  say,  33 

times  11  are  363,  and  14  are  377,  Ans.       ^^•'^'^ 

3.  What  is  the  cost  of  17  doz.  and  9  eggs  at 
12^  cts.  per  doz.? 

Here  ^  of  17  to  the  nearest  unit  is  9,       17/^ 
and  ^  of  12  is  9 ;.  then  nine  plus  9  is  18,       12^ 
BO  we  simply  say,  12  times  17  are  204  and  

18  are  222,  the  answer.  ^^'^^ 

4.  What  will  be  the  cost  of  15|  yds.  calico  at 
12^  cts.  per  yd.?  Ans.  $1.97. 


•48        ORTON  &  Sadler's  calculator. 

Where  the  Multiplier  is  an  Aliquot  Part  of  100. 

Merchants  in  selling  goods  generally  make  the 
price  of  an  article  some  aliquot  part  of  100,  as  in 
selling  sugar  at  12A  cents  a  pound  or  8  pounds 
for  1  dollar,  or  in  selling  calico  for  16J  cents  a 
yard  or  6  yards  for  1  dollar,  etc.  And  to  be- 
come familiar  with  all  the  aliquot  parts  of  100,  so 
that  you  can  apply  them  readily  when  occasion 
requires,  is  perhaps  the  most  useful,  and,  at  the 
same  time,  one  of  the  easiest  arrived  at  of  all  the 
computations  the  accountant  must  perform  in  the 
practical  calculations  of  the  counting-room. 

TABLE  OF  THE  ALIQUOT  PARTS  OF  100  AND  1000 
N.  B.  Most  of  these  are  used  in  business. 


m 

is  J  part  of  lOO 

H  is  A  part  of    100. 

25 

is  f  or  i  of  100. 

16f  is  ^2^  or  J  of    100. 

37i 

is  f   part  of  lOa 

331  is  i4  or  I  of    100. 

50 

is   f  or  J  of  100. 

66|  is  T^  or  1  of    100. 

62i 

is   I   part  of  100. 

83i  is  {^  or  t  of    100. 

75 

is  f  or  }  of  100. 

125    is  J  part  of  lOOO 

87J 

is   I  part  of  100. 

250    is  f  or  J  of  1000. 

6i 

is  ^  part  of  100. 

375    is  J  part  of  1000. 

18} 

is  ^  part  of  100. 

625    is  f  part  of  1000. 

31J 

is  A  part  of  lOO 

875    is   I  part  of  1000. 

To  multiply  by  an  aliquot  part  of  100, 

Rule  — Add  two  ciphers  to  the  multiplicand,  then 

take  scich  part  of  it  as  the  multiplier  is  part  of  100. 
N  B.  If  the  multiplicand  is  a  mixed  number  reduc< 

the  fraction  to  a  decimal  of  two  places  bofore  diyiding. 


COUNTmG-EOOM   EXERCISES. 


Examples. — 1.  Multiply  424  by  25. 

As  25  =  ^  of  100,  divide  42400  by  4  =  10600. 

N.  B.  If  the  multiplicand  is  a  mixed    number,  reduce  the 
reaction  to  a  decimal  of  two  places  before  dividing. 

2.  Give  the  cost  of  12^  yds.  cloth  @  18|c.  per  yd. 
Process. — 12^  =  ^]  changing  18|  to  a  decimal, 

we  have  18.75  -r-  8  =  $2.34f. 

Note. — Aliquot  parts  may  be  conveniently  used  when  the  mul 
tiplier  is  but  little  more  or  less  than  an  aliquot  part. 

3.  Multiply  24  by  11%. 

1st.  Multiply  24  by  16f  (the  one-sixth  of  100) 
Thus    24    X    16f    =  2400  ^  6  =  400 
As  nf  =  16f  +  1  multiply  24  by  1  =_24 
Hence  24  x  17f  =  the  two  products,  424 
^  49 


60        ORTON  &  Sadler's  calculator. 

3.  To  multiply  any  number  by  125  add  three 
ciphers,  and  divide  by  8. 

Multiply  3467  by  125.        Product,  433375. 

8)3467000 
433375 

Note.— By  annexing  three  ciphers  the  number  is  in- 
creased one  thousand  times;  and  by  dividing  by  8,  the 
quotient  will  be  only  one-eighth  of  1000,  that  is  125  times. 

4.  To  multiply  any  number  by  161  add  two 
ciphers,  and  divide  by  6. 

Multiply  3768  by  16?.  Product,  62800. 

6)376800 
62800 

5.  To  multiply  any  number  by  1661  add  three 
ciphers,  and  divide  by  6. 

Multiply  7875  by  166 1.     Product,  1312500. 

6)7875000 
1312500 

6.  To  multiply  any  number  by  83 J  add  two 
ciphers,  and  divide  by  3. 

Multiply  9879  by  33J.        Product,  32930C. 

8)987900 
829300    . 


COUNTING-ROOM   EXERCISES.  51 

Rationale. — As  in  the  last  case,  by  annexing 
two  ciphers,  we  increase  the  multiplicand  one  hun- 
dred times ;  and  by  dividing  the  number  by  3,  we 
only  increase  the  multiplicand  thirty-three  and 
one-third  times,  because  33J  is  one-third  of  100. 

4.  To  multiply  any  number  by  333J  add  three 
ciphers,  and  divide  by  3. 

Multiply  4797  by  333J.      Product,  1599000. 

3)4797000 

1599000 

5.  To  multiply  any  number  by  6§  add  two  ci- 
phers, and  divide  by  15  j  or  add  one  cipher  and 
multiply  by  f . 

Multiply  1566  by  6f . 

15)156600  ^'^«*— ^ 

10440  First  method. 

15660 
2 

3)31320 

10440  Second  method. 

6.  To  multiply  any  nunber  by  66f  add  three 
ciphers,  and  divide  by  15 ;  or  add  two  ciphers  and 
multiply  by  f . 


52    ORTON  &  Sadler's  calculator. 

Multiply  3663  by  66^. 

15)3663000 

244200  First  method. 

366300 
2 

3)732600 

244200  Second  method. 

7.  To  multiply  any  number  by  8J  add  two  ci- 
phers, and  divide  by  12. 

Multiply  2889  by  8J.     Product,  24075. 
12)288900 

24075 

8.  To  multiply  any  number  by  83 J  add  thre€ 
ciphers,  and  divide  by  12. 

Multiply  7695  by  83J.     Product,  641250. 
12)7695000 


641250 
9.     To  multiply  any  number  by  6 J  add  t\i» 
phers,  and  divide  by  16  or  its  f£.ctors — 4X4. 
Multiply  7696  by  6^.     Product,  48100. 
4)769600 

4)192400 

48100 


DIVISION   OF    FRACTIONS, 

WITH   ANALYSIS. 

As  the  base  of  all  numbers,  whether  whole  or 
fractional,  are  of  the  same  value,  inverting  any 
number  simply  demonstrates  the  number  of 
times  it  is  contained  in  a  single  unit. 

As  the  unit  takes  the  place  of  the  number, 
so  must  the  number  take  the  'place  of  the^iinit. 

Example  1. — Divide  6  by  7.     Ans.  f. 

By  inverting  the  divisor  we  find  j,  j;  now  if  7 
is  contained  in  one  unit  ^  of  one  time,  it  is  con- 
tained in  6,  six  times  ;^,  or  ^  Ans, 

Example  2. — Divide  5  by  ^.     Ans,  61. 

I  is  contained  in  a  unit  |  times,  therefore  it  is 
contained  in  f ;  JX4  =  V»  ^^  ^^  ^'^^• 
53 


54        orton  &  Sadler's  calculator. 

Example  3. — Divide  8 J  by  I.    A7is.  Hi. 
By  inverting  f  we  have  .j,  the  number  of  times 
it  is  contained  in  a  unit ;  therefore  it  is  contained 
8i,  8J  times  i  or  y  X|  =  %®  or  lli=llj  Ans. 
(Note.  8iX|  =  V^oi'lli-) 

Example  4. — Divide  6}  by  3.    A7is.  2i. 
As  3  is  contained  in  a  unit  J  of  one  time,  61  is 
contained  62  times  J,  or  thus, 

y  -^-3=  y  X  J = V  or  2i  ^m. 

Rule. — To  ascertain  the  number  of  times  the 
divisor  is  co7itained  in  a  unit,  invert  the  divisor 
and  multiply  by  the  units  in  the  dividend. 

Example  5. — Divide  3  by  4.     Ans.  5i. 

Solution. — 4  ^^  contained  in  a  unit  |  times,  there- 
fore it  is  contained  in  3,  3  times  |  or  ^5'  =5i  Ans. 

Note. — Inverting  the  divisor  shows  how  often  it  is 
contiiined  in  a  unit,  and  multii)lying  this  number  by  the 
dividend  gives  the  quotient  of  all  examples  in 

DWISION   OF  FRACTIONS. 


Example  6. — Divide  \  by  i.     Ans. 


Solution. — }  is  contained  in  a  unit  4  times, 
therefore  it  is  contained  in  |,  1  of  4  times,  or 
I  times  Ans. 

Example  7. — Divide  f  of  f  by  |  of  J. 
Statement,  J  of  |_  3 

Note. — Simply  reject  or  cancel  factors  in  the  dividend 
and  divisor,  tJiaa  you  Juive  i  as  the  quotient. 


DIVISION   OF   FRACTIONS.  55 

Example  8.— Divide  J  of  f  of  3  by  ^^  of  |  of  §. 

1.^  Statement-^  of  ?  of  f  |  Rejecting  common  fac- 
304  n^  >  toi-s  and  we  have  lor 
^^  ^^  ^  "^^  0  the  result  f  for  23^2. 


2d  Staieme-it. — 

$  fi  a 

By  boxiug  the 
deuominators. 

43^ 

%  ^  i. 

=  f|or2Jj^ns. 

i0     0     5 
5 

Rule. — Beject  or  cancel  factors  common  to  the 

divisor  and   dividend.      Multiply  the  remaining 

terms  between  lines  (or  inside  the  box)  together 

for  the  desired  denominator;  and  the  terms  above 

and  below  the  lines  (or  outside  the  box)  for  the 

desired  numerator. 

Note.— The  above  rule  is  commended  as  the  most  sim- 
ple one  devised  for  the  division  of  fractions. 

Example  9.— Divide  i  of  f  by  %  of  i. 

The  application  of  the  above  rule  to  Example 
9  produces  the  following  statement,  so  simple 
that  it  can  readily  be  understood  by  the  most 
ordinary  pupil. 

Draw  a  figure  of  four  sides,  representing  a 
box,  then  write  the  example  as  follows ; 
3        2 


y_Tj==§8ort 


56   ORTON  &  Sadler's  calculator. 

The  numerator  of  the  fraction  will  be  found 
outside  the  box,  which,  multiplied  together,  will 
give  the  desired  numerator,  thus, 
3X2X5X2  =  60 

The  denominator  will  be  found  upon 
the  inside  of  the  box,  which,  multiplied 
together,  will  give  the  desired  denomi- 
nator, thus,  4X5X4X1  =  80 

Note.  When  factors  are  common  to  each  other  always 
take  advantage  of  cancellation. 

An  investigation  of  our  new  method  of  treating 
the  division  of  fractions  will  prove,  no  doubt,  to  be 
the  most  simple  and  practical  ever  yet  devised. 
MULITIPLICATION    AND    DIVISION. 

To  multiply  i,  is  to  take  the  multiplicand  J  of 
one  time;  that  is,  take  i  of  it,  or  divide  it  by  2. 

To  multiply  by  J,  take  a  third  of  the  multipli- 
cand, that  is,  divide  it  by  3. 

To  multiply  by  f,  take  i  first,  and  multiply 
that  by  2;  or,  multiply  by  2  first,  and  divide  the 
product  by  3. 

Example. — If  1  cord   of  wood   cost   $6.00, 
what  will  be  the  cost  of  i  of  a  cord  ? 
6^otoio/i.— $6.00  X 1= $4.50 

3 

1800-r-4=  4.50 
What  will  j  of  a  cord  cost  ? 

jXi=iH-f=$2.00 


MULTIPLICATION   AND   DIVISION.  57 

EXAMPLES. 

I.  What  will  360  barrels  of  flour  come  to  ai  5| 

aoUars  a  barrel.  At  1  dollar  a  barrel  it  would  be  360 

dollars ;  at  5 J  dollars,  it  would  be  5 J  times  as  much. 

360 

5  times,  1800 

J  of  a  time,        90 

Ans.  $1890 

Before  we  attempt  to  divide  by  a  mixed  number, 
such  as  2^,  3J,  5f ,  etc.,  we  must  explain,  or  rather 
observe  the  principle  of  division,  namely:  That 
the  quotient  will  he  the  same  if  we  multiply  the  divi- 
dend and  divisor  hy  the  same  number.  Thus  24 
divided  by  8,  gives  three  for  a  quotient.  Now,  if 
we  double  24  and  8,  or  multiply  them  by  any  num- 
ber wnatever,  ana  then  divide,  we  shall  still  have  3 
for  a  quotient.     16)48(3;  32)96(3,  etc. 

Now,  suppose  we  have  22  to  be  divided  by  5^ ; 
we  may  double  both  these  numbers,  and  thus  be 
clear  of  the  fraction,  and  have  the  same  quotient. 
5^)22(4  is  the  same  as  11)44(4. 

How  many  times  is  \\  contained  in  12?  Ans, 
J  usi  as  many  times  as  5  is  contained  in  48.  The 
5  is  4  times  1  J,  and  48  is  4  times  12.  From  these 
observations,  we  draw  the  following  rule  for  divid- 
ing by  a  mixed  number. 


58        ORTON  &  Sadler's  calculator. 

EuLE. — Multiply  the  whole  number  by  the  lower 
term  of  the  fraction ;  add  the  upper  term  to  the  pro- 
duct for  a  divisor ;  then  multiply  the  dividend  by 
the  lower  term  of  the  fraction^  and  then  divide. 

How  many  times  is  1^  contained  in  36?  An», 
30  times. 

N.  B.  If  we  multiply  both  these  numbers  by  5, 
they  will  have  the  same  relation  as  before,  and  a 
quotient  is  nothing  but  a  relation  between  two 
numbers.  After  multiplication,  the  numbers  may 
be  considered  as  having  the  denomination  of  fifths. 

How  many  times  is  J  contained  in  12  ?  Ans.  48 
times. 

One-fourth  multiplied  by  4,  gives  1 ;  12,  multi- 
plied by  4,  gives  48.  Now,  1  in  48  is  contained  48 
times. 

Divide  132  by  2|.  Ans.  48. 

Divide  121  by  15J.  Ans.    8 

How  many  times  is  |  contained  in  3  ?  Ans.  4 
times. 

By  a  little  attention  to  the  relation  of  numbers, 
we  may  often  contract  operations  in  multiplication. 
A  dead  uniformity  of  operation  in  all  cases  indi- 
cates a  mechanical  and  not  a  scientific  knowledge 
of  numbers.  As  a  uniform  principle,  it  is  much 
easier  to  multiply  by  the  small  numbers,  2,  3,  4, 
5,  tlian  by  7,  8,  9. 


PERGENTA^GE. 


The  greater  portion  of  all  arithmetical  calcu- 
lations, as  applied  to  every-day  business  transac- 
tions, being  based  upon  percentage,  it  is  important 
that  the  foregoing  principles  and  illustrations  be 
thoroughly  mastered. 

Percentage  is  the  process  of  computing  by  the 
hundred. 

Per  cent,  or  rate  per  cent,  means  by  the  hun- 
dred, and  is  represented  by  the  character  %  in- 
stead of  being  written  thus,  5%,  20%,  100%. 
Any  %  less  than  1%,  can  be  written  in  the  form 
of  a  fraction,  thus,  }%,  f  %,  or  expressed  deci- 
mally, thus,  .002%,  .0025%. 

THE  FIVE  FACTS 

To  be  considered  in  percentage  are : 

Is^.  The  Base,  2d,  The  Rate.  M,  The  Per- 
centage,    4th.  The  Amount.     5th.  The  Difference. 

The  Base — Is  the  number  upon  ^vhich  the 
Percentage  is  calculated. 

The  Rate — Is  the  number  denoting   the  per 
cent,  (or  hundredths)  of  the  Base  taken,  and  is 
always  used  as  the  multiplier. 
59 


» 
o 

» 

Pi 

o 

H 

3 

P4 

Ai 


PERCENTAGE.  61 

The  Percentage — Is  the  sum  (in  hundredths) 
obtained  from  multiplying  the  Base  by  the  Bate, 

The  Amount — Is  the  Base  increased  by  add- 
ing the  Percentage. 

The  Difference — Is  the  Base  diminished  by 
subtracting  the  Percentage, 

APPLICATION  OF  PERCENTAGE. 
Given — The  Base  and  Bate, 
To  Find — T'he  Percentage. 
Rule  I. — Multiply  the  Base  by  the  Rate,  ex- 
pressed decimally^  and  point  off  two  places  from 

the  right 

PROBLEM. 

What  is  9%  (or  the  percentage)  of  $800?— 

jy  Base,  $800.  Multiply  by  the  rate. 

rrocess.  Rate,      .09 

BXR=  Percentage,  $72.00— Point  off  two  figures 
from  the  right,  and  the  result  is  the  per  cent.  $72. 

Given — The  Base  and  Bate. 

To  Find — The  Amount. 

Rule  II. —  To  the  Base  add  the  Percentage. 

PROBLEM. 
What  is  the  amount  of  $800,  increased  9%  ? 
p.  Base,  $800.  Plus  the  Percentage. 

Percentage,     72.  Obtained  under  Rule  I. 

B+P=  Amount,  $872. 
6 


62        ORTON  &  Sadler's  calculator 

Given — The  Base  and  Rate, 
To  Find — The  Difference, 
Rule  III. — From  the  Base  subtract  the  Per- 
centage. 

PROBLEM. 

What  is  the  Difference  (or  proceeds)  of  $800, 
less  9%? 

p  Base,  $800.  Minus  the  Percentage. 

Percentage,     72.  Obtained  under  Rule  I. 

B—P=  Difference,   $728.— or  Proceeds. 


Given — The  Base  and  Percentage. 

To  Find— TAe  Rate, 

Rule  IV. — Divide  the  Percentage  by  the 

Base,  expressed  decimally. 

Note. — When  cent*  ^ire  not  shown  in  the  percentage 
add  two  ciphers. 

PROBLEM. 

Bought  of  W.  H.  Sadler,  invoice  of  Orton's 
Lightning  Calculators,  for  $850,  and  sold  them  at  a 
profit  of  $297.50— what  %  did  I  make  ?  Ans.  35%. 

Process, — Percentage  divided  by  the  Base. 
Base,  $850.    Percentage,  $297.50.    P-J-B==Rate. 

Base.  Percentage.  Rate. 

-850)297.50(35% 
2550 

4250 
4250 


PERCENTAGE.  63 

Given — Rate  and  Percentage, 

To  Find— TAe  Base, 

Rule  V. — Divide  the  Percentage  by  the 
Rate. 

Note. — When  cents  are  not  shown  in  the  percentage 
annex  two  ciphers. 

PROBLEM. 

Sold  William  Callen,  Jr.,  invoice  of  Orton'a 
LightDing  Calculators,  upon  which  I  gained 
$297.50.  Ascertaining  my  profit  to  be  35% — 
what  was  the  amount  or  cost?     Ans.  $850. 

Process. — Percentage  divided  by  the  rate. 
Percentage,  $297.50.    Rate,  35%.    P-T-R=Base. 

Bate.  Percentage.  Base. 

35)297.50(850. 
280 


Given — Amount  and  Rate, 
To  Find— T/ie  Base, 

Rule  VI. — Divide  the  Amount  by  100, 
added  to  the  Rate. 

Note. — When  cents  are  not  shown  in  the  amount  annex 
two  ciphers. 

PROBLEM. 

Sold  John  G.  Scouten,  invoice  of  Orton^s 
Lightning  Calculators,  amounting  to  $1147.50, 
and  made  a  profit  of  35%.  What  did  the  books 
cost?    Ans,  $850. 


64        ORTON  &  Sadler's  calculator. 

Process. — Amount  divided  by  100  plus  the 
rate. 

Amount,  $1147.50.  Rate,  35%.  100  in- 
creased by  the  rate  35=1.35. 

Rate.      Amount.      Base  or  Cost. 

1.35)1147.50(850 
1080 
675 
675 


A-T-100+R=Base.  0 


Given — The  Difference  and  Bate, 
To  Find— T/ie  Base. 

EuLE  VII. — Divide  the  Amount  by  100  less 
the  Rate. 

Note. — When  cents  are  not  shown  in  the  difference 
annex  two  ciphers. 

PROBLEM. 

Invoice  of  Orton's  Lightning  Calculators  sold 
William  David,  were  damaged  by  water,  and  he 
was  compelled  to  sell  them  for  $552.50,  thereby 
losing  35%.     What  did  they  cost?     Ans.  $850. 

Process. — Difference  divided  by  100  less  the 
rate. 

Difference,  $552.50.  Rate,  35%.  100  less  the 
r.ite  35=65. 

Bate.  Difference.  Base  or  Cost. 

65)552.50(850. 
520 

325 
325 
Dh-1.00— R=Base.    q 


PROFIT  AND  LOSS 
Are  terms  denoting  the  gain  or  loss  arising  from 
business  transactions. 

The  preceding  Rules  under  percentage  are 
specially  adapted  to  the  majority  of  business 
transactions ;  we  therefore  call  attention  to  their 
application  : 

Capital  or  Cod  is  treated  as  the  Base. 

Per  cent  (%)  of  profit  or  loss  is  treated  as 
the  Rats. 

Sum  Gained  or  Lost  is  treated  as  the  Per- 
centage. 

Selling  Price  is  treated  as  the  Amount. 

Cost,  less  the  Loss,  is  treated  as  the  Difference. 
65 


ee 


ORTON    &   SADLER  S    CALCULATOR. 


SHORT  METHODS  in  MERCHANDISING. 

When  the  Bate  is  an  Aliquot  part  of  $1.00  or 
100,  instead  of  following  Hule  I.  of  Percentage 
the  labor  will  be  greatly  abridged  by  applying 
the  short  method,  as  explained  on  page  48. 

Note. — Aliquot  parts  of  a  number  are  such  whole  or 
mixed  numbers  as  will  divide  it  without  a  remainder. 
Thus  2,  2^,  3^,  and  5  are  aliquot  parts  of  10,  being  con- 
tained in  it  5,  4,  3,  and  2  times. 

TABLE  OF  ALIQUOT  PARTS. 


100 

ITon 

1  ft. 

Aliquot  parts  of 

1 

10 

or 

1000 

of 

or 

1  A. 

~14 

5 

$1.00 

2000  lb. 

1  doz. 

One  half  is.... 

60 

500 

1000 

6 

80  sq. rd. 

One  third  is.   . 

v. 

'i% 

33><J 

3331^ 

6662^ 

4 

One  fourth  is. . 

W 

Wi 

25 

250 

500 

3 

40  sq.  rd. 

One  fifth  is. . . . 

\ 

2 

20 

200 

400 

32      " 

One  sixth  is.   . 

% 

IK 

10'^ 

166:^ 

333>g 

2 

One  eighth  is. 

% 

iVi 

12K 

125 

250 

20  sq.  rd. 

One  tenth  is  . . 

A 

1 

10 

100 

200 

16      " 

One  twelfth  is 

.\ 

Wz 

83}^ 

I 

etc. 

Example.— Multiply  843  X  83}. 

Proces.?.— Since  83i  is  ,V  of  1000,  83}  times 
any  number  is  j^^  ^^  1000  times  that  number. 
Therefore,  to  multiply  843  by  83}  we  simply 
multiply  843  by  1000,  and  divide  the  product  by 
12,  the  quotient  will  be  the  required  product  thus: 

843  X  1000  =  843G00  -i- 12  =  70250. 


Is  the  sura  paid  for  the  use  of  money. 

Simple  Interest  is  interest  on  the  principal  only. 

Annual  Interest  is  simple  interest  on  the  prin- 
cipal, and  on  each  year's  interest  from  the  date 
of  its  accruing  until  paid. 

Compound  Interest  is  interest  allowed  on  inter- 
est and  principal  combined. 

^^^  Calculations  in  compound  interest  may  be  abridged 
by  use  of  tables  on  page  209. 

Note. — Interest  may  be  compounded  and  added  to  the 
principal  annually,  semi-annually,  or  quarterly,  as  per 
agreement  between  lender  and  borrower. 

Accurate  Interest  is  interest  calculated  on  the 
basis  of  365  days  to  the  year.  It  is  reckoned 
by  the  usual  methods,  and  ^^^  of  the  sum  de- 
ducted, except  in  case  of  leap  year  when  g^j-  of 
the  interest  is  subtracted. 

Legal  Interest  is  the  rate  fixed  according  to  law. 

Usury  is  when  a  higher  rate  of  interest  is 
paid  than  is  sanctioned  by  law. 

Note.— See  pages  297-298. 

The  Principal  is  the  sum  in  use  and  upon 
which  interest  is  paid. 

*  For  a  better  understanding  of  the  practical  calcula- 
tions of  interest,  the  author  refers  to  other  portions  of  this 
work. 

67 


^S       ORTON  &  Sadler's  calculator. 

The  Bate  of  Interest  is  the  price  paid  for  the 
use  of  one  dollar. 

The  Amoxint  is  the  principal  with  the  accrued 
interest  added. 

As  in  Percentage  there  are  five  facts  to  be 
considered,  viz. : 

Principal^  Rate  per  annum,  Literest,  Time 
and  Amount. 

APPLICATION  OF  PERCENTAGE. 

The  Principal  is  treated  as  the  Base, 

The  Pate,  or  price  paid  per  annum,  is  treated 
as  the  Pate. 

The  Interest  is  treated  as  the  Percentage. 

The  Principal  and  Interest  is  treated  as  the 
Amount. 

The  Time  is  an  additional  element  in  Interest. 

Given — Principal,  RatCy  and  Time  (in  days). 

To  Find — The  Interest  at  any  rate  per  cent. 

EuLE  I. — Time  in  days.  Multiply  the  Prin- 
cipal hy  the  Rate,  and  the  product  by  the  time 
{expressed  in  days) ;  then  divide  the  result  hy  36 
and  the  quotient  will  he  the  Interest  in  millsy 
or  Tij^oo  oj  $1.00.       

Given — Principal,  Rate,  and  Time  (in  months). 
To  Find — The  Interest  at  any  rate  per  cent. 


INTEREST.  69 

KuLE  II. — Time  in  months.  Multiply  the 
Principal  by  the  Rate,  and  the  product  by  the 
Time  {number  of  months);  divide  the  result  by 
12,  and  the  quotient  will  he  the  Interest  in 
cents,  or  yj^  of  $1.00. 

GiYB^— Principal,  Mate,  and  Time  (in  yeare). 

To  Find — The  Interest  at  any  rate  per  cent. 

Rule   III. — Time    in    years.      Multiply  the 

Principal  by  the  Rate,  aiid  the  product  by  the 

Time  {number  of  years),  the  result  will  be  the 

Interest  in  cents,  or  yjo  of  $1.00. 

Note. — The  above  rules  are  not  specially  recoraraended 
for  general  business  use,  but  are  here  presented  to  call 
attention  to  the  principle  (percentage)  upon  which  all  in- 
terest calculations  are  based. 

For  Short  Methods  and   Practical  every-day 

rules  the  author  refers  to  the  portion  of  this 

work  devoted  exclusively  to  interest  calculations. 

Given — The  Principal,  Rate,  and  Time, 

To  Find — The  Interest  and  Amount. 

Rule. — Calculate  the  Interest  for  the  Time 

at  the  stated  Rate,  and  add  to  the  Principal, 

the  product  will  be  the  Amount. 

Example. — What  will  $1000  amount  to,  in- 
vested for  8  months  at  7%  interest? 
PXRXT-r-12  =  Interest. 
Principal,  $1000  +  46.67  =  $1046.67  Amount. 


70        ORTON  &  Sadler's  calculator. 

Process.— Principal,  1000  X  Rate,  7%. 

07 

7000  X  Time  in  months. 
8 


12)56000  Product-f-12=Interest 
$46,666  [S46.67. 


Given — The  Principal,  Interest,  and  Time, 

To  Fi:sT>— The  Bate, 

Rule. — Divide  the  stated  Interest  by  the  in- 
terest on  the  Principal,  for  the  Time  calctdated 
at  1%  per  annum,  the  quotient  will  be  the  Rate. 

If  $5000,  invested  for  1  year  and  6  months, 
gains  $525 — what  is  the  rate?    Ans,7%. 

Proem.— Principal,  $5000  X  Rate,  1J%,  for 
1  year  and  6  months =$75.  Interest  at  1% — 
Stated  Interest  divided  by  $75=  the  Rate. 

stated  Int. 

Interest  for  time  @  1%=$75)525(7%  Rate. 

525 


Given — The  Bate,  Time,  and  Interest 

To  Find — The  Principal 

Rule. — Divide  the  stated  Interest  by  the  in- 
terest on  one  dollar  for  the  stated  Time,  at  the 
dated  Rate. 

Note. — When  cents  are  shown  in  the  interest  annex 
two,  and  for  mills  three  ciphers. 


INTEREST.  71 

Example. — What  principal  will  gain  $525 
interest  in  1  year  and  6  months,  at  7%. 

Process. — Interest  on  $1.00  for  1  year  and  6 
months,  is  .105,  or  lOJ  cents. 

stated  Int. 

Interest  on  $1.00=.105)525000(5000  Principal. 
525 


000 


Given — The  Bate,  Time,  and  Amount 

To  Find — The  Principal, 

Rule. — Dioide  the  Amount  by  100,  plus  the 

interest  on  one  dollar  for  the  stated  Time,  at  the 

stated  Rate. 

Note. — When  cents  are  shown  in  the  interest  annex 
two,  and  for  mills  three  ciphers. 

Example. — What  principal  will  amount  to 
$5525,  in  1  year  and  6  months,  at  7%  ? 

Process. — Interest  on  $1.00  for  1  year  and  6 
months,  at  7%  =.105,  or  10}  cents. 

100+. 105=  1.105)$525.000(5000  Principal. 
525 

f^'  000 

Given — Principal,  Bate,  and  Interest. 

To  Find— T/ie  Tiine. 

Rule. — Divide  the  stated  Interest  by  the 
interest  on  the  Pkincipal,  for  one  year,  at  the 
stated  Rate. 


72   ORTON  Sc  Sadler's  calculator. 

Note. — The  integer  or  whole  number  in  the  quotient 
will  be  ihe  time  in  years.  For  months,  multiply  the  deci- 
mal or  remainder  by  12,  and  divide  as  before,  the  quotient 
will  be  the  time  in  months.  For  days,  multiply  the  deci- 
mal or  remainder  in  months  by  30,  and  divide  again,  the 
quotient  will  be  the  time  in  days. 

Example. — In  what  time  will  $5000  amount 
to  $5455? 

Process. — Principal,  $5000.  Interest  1  year 
at  6%  =$300. 

Stated  Interest,  $455-r-300=Time,  1  year,  6 
mouths,  6  days. 

300)455(1  year. 
300 
'155 
12 
300)1860(6  months. 
1800 
60 
30 


300)1800,6  days. 
1800 

TEST   EXAMPLE. 

In  what  time  will  $3000,  at  7%,  amount  to 
$3570.50?     Ans.  2  years,  8  months,  18  days. 

Note. — When  cents  are  shown  in  the  stated  interest 
annex  two  ciphers  to  the  interest  on  the  principal  for  one 
year,  providing  cents  are  not  shown,  and  vice  versa. 


>ii--i>i*ni.'Lj; 


Is  the  percentage  off  or  allowance  made  for  the 
payment  of  money  before  maturity. 

COMMERCIAL  DISCOUNT. 

In  this  connection  the  term  discount  is  used 
without  reference  to  time.  It  is  the  sura  or  per- 
centage deducted  from  the  Lid  or  asking  price 
of  goods. 

It  is  the  allowance  or  deductions  made  from 
Invoices  or  Bills  purchased,  in  consideration  for 
prompt  or  ca^h  payment. 

Note. — Certain  goods  usually  sold  on  credit  may  be 
bought  for  less  price,  providing  cash  settlements  are  made. 
The  sum  or  abatement  from  the  credit  price  or  terms, 
such  as,  2i,  5,  or  6%  off,  is  termed  discount. 

Again,  on  various  classes  of  articles  the  retail  price  is 
fixed  by  the  publisher  or  manufacturer,  and  certain  de- 
ductions are  allowed  to  importers  or  wholesale  buyers, 
which  is  given  in  the  form  of  a  per  cent,  off,  such  as,  25, 33i, 
and  40%,  with  further  allowances  for  Net  Cash  payment. 

Net  Price  of  an  article  is  the  selling  or  asking 
price,  less  the  discount. 

Net  Proceeds,  or  cash  value  of  a  bill,  is  its  face 
with  the  discount  deducted. 

APPLICATION  OF  THE  RULE  of  PERCENTAGE 
Base — The  selling  price  or  face  of  bill. 
Pate — The  rate  per  cent,  of  deduction. 
Percentage — The  discount   or  amount  of  de- 
duction.     -  rjQ 


74        ORTON  &  Sadler's  calculator. 

Rule  I. — Multiply  the  selling  price  or  face  of 
the  bill  by  the  rate  per  cent,  of  dedxiction,  and  the 
product  will  be  the  Commercial  discount 

Rule  II. — From  the  selling  price  or  face  of  the 
bill  deduct  the  commercial  discount,  and  the  differ- 
ence will  be  the  Net  Price,  Cash  Value,  or  Net 
Proceeds. 

Note. — The  practical  application  of  the  above  rules 
has,  previously,  been  so  fully  illustrated  that  examples 
here  are  not  deemed  necessary. 

TKUE  DISCOUNT 

Is  the  difference  between  the  face  of  the  debt 

and  its  present  worth  or  value,  therefore  it  is 

evidently  the  interest  on  the  present  worth  from 

date  to  the  time  of  maturity. 

Note. — Every  debt  or  note  due  at  some  future  time, 
without  interest,  has  some  existing  value  now,  and  that 
value  is  termed  Present  Worth ;  therefore, 

Present  Worth  is  such  a  sum  as  being  placed 

at  interest    to   the    date  of   maturity   as   will 

amount  to  the  stated  debt. 

APPLICATION  OF  PERCENTAGE. 

The  Present  Worth  is  treated  as  the  Base. 
The  Debt  or  Face  of  Bill  is  treated  as  the 
Amount 

The  True  Discount  is  treated  as  the  Difference. 


PERCENTAGE.  75 

TO  ASCERTAIN  THE  PRESENT  WORTH. 

Rule. — Divide  the  amount  of  the  debt  or  face 
of  bill  by  100  plus  the  interest  on  $1.00  for  the 
given  time,  at  the  stated  rate. 

Note. — When  cents  are  shown  in  the  interest  annex 
two,  for  mills  three,  ciphers,  to  the  debt  or  bill. 

TO  ASCERTAIN  THE  TRUE  DISCOUNT. 

Rule. — From  the  debt  or  face  of  bill  subtract 
the  present  worth. 

Example. — What  is  the  present  worth  of 
S618— Note  due  in  6  mouths,  at  6%  ? 

Process.— Int.  on  $1.00  6  mos.  @  6%  is  .03 
.03+100=1.03)618.00(600.— Present  worth. 
618_ 

00 
618-600=18.— True  discount. 


Proo/.— 1.00X600=$ 

618—600=     18. 


Face  of  note,  $618. 


76       ORTON  &  Sadler's  calculator. 

BANK  DISCOUNT 
Is  the  Interest  paid  in  advance,  or  deducted  from 
the  face  of  a  note  or  time  draft. 

Note. — Should  the  paper  offered  for  discount  bear  inter- 
est, bank  discount  is  the  interest  on  the  amount  due  at 
maturity  instead  of  on  the  face.  In  discounting,  the  time 
is  reckoned  by  days,  and  the  basis  of  calculation  360  days 
to  the  year.  In  discounting  paper,  banks  include  the 
day  on  which  the  note  is  discounted  and  the  day  on  which 
it  matures. 

Discount  is  the  sum  deducted  from  the  face  of 
a  note  or  acceptance,  which  is  the  interest  for  the 
number  of  days  from  date  of  discount  to  maturity. 

Proceeds  is  the  sum  given  or  amount  of  the 
note  or  acceptance,  discounted,  less  the  interest. 

Maturity  of  a  note  is  the  time  or  date  it  be- 
comes due,. including  days  of  grace. 

Days  of  Grace  are  the  three  days  allowed  by 
law  for  payment,  after  the  expiration  of  the  time 
specified  in  the  note. 

Protest  is  the  formal  legal  notice  made  by  a 
Notary  Public,  notifying  the  maker  and  endorsers 
of  the  non-acceptance  or  payment  of  paper  for 
which  they  are  held  liable. 

Note. — A  protest  for  non-payment  must  be  made  on 
the  last  of  three  days  of  grace,  unless  that  day  should 
occur  on  Sunday,  or  legally  authorized  holiday,  in  which 
case  protest  must  be  made  on  the  day  previous. 

Non-Protest. — In  case  of  non-protest  wherein 
there  are  endorsers  to  commercial  paper,  they  are 
legally  released,  and  the  holder  can  only  look  to 
the  maker  for  payment. 

Protest  Waived. — Consent  of  drawei*s  or  en- 
dorsers to  hold  themselves  responsible  for  pay- 
ment without  the  necessity  of  protest. 


COMMISSION. 


The  rate  of  commission  or  brokerage  in  gene- 
rality of  cases  is  established  by  custom,  ranging 
from  i%  to  1%.  A  commission  merchant 
generally  gets  2}%  for  selling,  and  an  additional 
2J%  if  he  guarantees  the  payment. 

Commission — Is  the  sum  paid  by  the  principal 
to  an  agent  for  selling  goodc  or  property,  or  col- 
lecting money. 

Consignment — Goods  sent  to  a  commission 
merchant  to  be  sold. 

Consignor — The  party  sending  the  goods,  or 
shipper. 

77 


78        ORTON  &  Sadler's  calculator. 

Consignee — The  party  to  whom  the  goods  are 
sent. 

Proceeds — The  sum  remaining  after  all  ex- 
penses are  paid. 

Account  Sales — Consignees'  written  statement 
to  the  consignor,  showing  at  what  price  the  goods 
were  sold,  the  expenses,  and  the  net  proceeds. 

Guarantee — Pledge  or  security  given  by  the 
commission  merchant  for  all  goods  sold  on  credit. 

Broker — One  who  sells  or  purchases  goods, 
stock,  etc.,  by  direction  of  another,  without  hav- 
ing them  in  his  possession. 

Brokerage — The  sum  paid  a  broker  for  his 
services. 

The  principles  and  workings  of  percentage 
involved  in  commission  and  brokerage  are  the 
same  as  those  heretofore  treated. 

CORKESPONDING  TERMS. 

The  Base  is  the  amount  of  sales,  investments, 
or  collecting. 

The  Rate  is  the  per  cent,  allowed  for  services. 
The  Percentage  is  the  Commission  or  Brokerage. 
The  Amount  or  Difference  is  the  Net  Proceeds. 

KuLE  I. — To  find  the  Commission  or  Broker- 
age,  multiply  the  Base  by  the  Rate. 
9 


COMMISSION.  79 

Rule  II. —  To  find  the  Rate^  divide  the  Com- 
mission  by  the  Base. 

Rule  III. — To  find  the  Base,  divide  the  Com- 
mission by  the  Bate. 

Note. — Wherein  a  certain  sum  is  supplied  a  broker  for 
investment  or  purchases,  from  which  the  pay  for  his  com- 
mission is  to  be  taken  ;  commission  on  his  own  commission 
not  allowed,  we  have  the  following 

Rule. — Divide  the  sum  supplied  by  100  plus 
the  rate  %  of  commission^  the  quotient  will  be  the 
Net  Proceeds;  this  sum  subtracted  from  the  Amount 
will  give  the  Commission. 

Example. — James  G.  Moulton  remits  a 
broker  $10,000  with  instructions  to  invest  in 
cotton,  his  commission,  2^%,  which  is  to  be  de- 
ducted— what  is  the  amount  of  cotton  pur- 
chased?    What  is  his  commission? 

Working  amount  to  be  invested,  =  100  % 
Commission  on  sum,    =      2} 
Total  on  purchase,         =  102i% 

Sum  furnished,  $10,000-r-1.025=$9756.10  Inv. 
$10,000-  9756.10=243.90  Commission. 
$9756.10 X. 025=  243.90  Commission. 

TEST   example. 

How  many  bushels  of  corn  can  be  purchased 
for  $3485 — the  market  price  being  68  cents  per 
bushel,  and  your  agent's  commission  for  pur- 
chasing 2i  %  ?    Ans.  5000  bushels. 


Is  a  contract  issued  by  companies,  wherein  they 
agree  for  a  certain  consideration  to  indemnify 
the  owner  or  holder  of  certain  property  against 
loss  or  damage  by  fire  or  shipwreck,  etc. 

The  Underwriter  is  the  Insurance  Agent  who 
acts  for  the  company. 

The  Insured  is  the  party  asking  for  protection, 
and  in  whose  favor  the  policy  is  issued. 

The  Policy  is  the  written  contract  issued  by 
the  company,  describing  the  property,  amount 
of  risk,  and  conditions  of  indemnity. 

The  Rate  or  per  cent,  of  Premium  is  the  cost 
of  $100  of  insurance. 

The  Premium  is  the  amount  paid  the  com- 
pany for  insurance,  and  is  generally  calculated 
at  a  certain  per  cent,  on  the  amount  of  insurance. 

PERCENTAGE  AS  APPLIED  TO  INSURANCE. 

The  Amount  is  treated  as  the  Base, 

The  per  cent,  of  Premium  is  treated  as  the  Rate, 

The  Premium  is  treated  as  the  Percentage. 

Given — Amount  of  Insurance  and  Rate. 

To  Find — The  Premium. 

Rule. — Amount  X  by  the  Rate  =  Premium, 

Given — Premium  and  Rate  of  Insurance, 

To  Find — The  Amount. 

Rule. — Premium  expressed  in  cents  -j-  by  the 
Rate  =  Amount. 

Given — Amount  and  Premium. 

To  FmD— The  Rate. 

Rule. — Premium  expressed  in  cents  -v-  by  the 
Amount  =  Rate. 

80 


INVESTMENTS. 


81 


INVESTMENTS. 

CAPITAL    AND    STOCKS. 

Capital  is  money  invested  in  business  or  private 
enterprises,  conducted  under  individual  or  co- 
partnership management. 

Capital  Stock  is  money  or  property  invested 
by  sundry  persons  in  manufacturing,  railroading, 
banking,  etc.,  and  is  generally  divided  into  cer- 
tificates or  shares  of  $100  each. 

The  management  of  such  enterprises  is  con- 
trolled by  a  Board  of  Directors,  from  among 
whose  number  executive  oflSicers  are  elected  or 
appointed. 


82   ORTON  &  Sadler's  calculator. 

Certificates  of  Stock  are  official  documents 
issued  by  the  corporation  or  company,  represent- 
ing a  certain  number  of  shares  of  the  joint  capi 
tal  to  which  the  holder  is  entitled. 

The  Par  Value  of  stocks  is  the  sura  or  nomi- 
nal value  for  which  they  were  issued,  as  expressed 
on  their  face. 

The  Market  Value  is  the  sum  for  which  they 
can  be  sold. 

Stocks  are  at  Par  when  their  market  value  is 
the  same  as  their  face. 

Stocks  are  below  Par  when  their  market  value 
is  less  than  their  face. 

Stocks  are  above  Par  when  their  market  value 
is  in  excess  of  their  face  value. 

Note. — Shares  representing  $100  each,  when  quoted  at 
$100,  are  worth  par ;  when  at  $110  or  over  $100  are  above 
par,  and  when  at  $90  or  less  than  $100  are  below  par. 

Market  quotations  of  stocks  are  generally  quoted  at  a 
certain  per  cent,  above  or  below  the  par  value. 

The  value  of  stocks  depend  upon  the  success  and  pros- 

Serity  of  the  business  they  represent,  and  per  cent  of 
ividend  declared. 

Assessment  is  the  sum  called  for  from  the 
stockholders  to  make  up  any  deficiency  or  losses 
that  may  arise  in  conducting  the  business. 

Dividend  is  the  sum  paid  the  stockholders, 
and  is  a  division  of  the  profits  of  the  company. 

Assessments  and  Dividends  are  calculated  upon 
a  certain  per  cent,  of  the  par  value  of  the  stock. 


INVESTMENTS.  Qo 

Brokerage. — The  party  buying  and  selling 
stocks  is  called  a  Broker  or  Stock  Jobber,  and 
the  compensation  received  for  his  services  is 
termed  Brokerage.  The  usual  rate  of  brokerage 
is  i  to  i  per  cent,  of  the  par  value  of  stock  pur- 
chased or  sold. 

PERCENTAGE. 
The  majority  of  business  transactions  that 
arise  in  connection  with  stocks  may  be  readily 
calculated  by  applying  the  principles  of  percent- 
age heretofore  shown,  as  an  examination  of  the 
following  illustrations  will  show : 

To  ascertain  the  Cost,  including  Brokerage. 

Rule. —  To  the  market  value  of  one  share  add 

the  brokerage^  and  multiply  by  the  number  of  shares. 

Example. — What  will  100  shares  of  Balti- 
more &  Ohio  Railroad  stock  cost,  market  value 
127S,  brokerage  i%? 

PROCESS. 

127^  (Cost  of  1  share)  +  }/^  (Brokerage)  =  12-7)^  Cost  of  1  sbar©. 
1273^X  100  (Number  of  ahare8,)  =  $l.:J,750  Cost. 


To  ascertain  the  number  of  Shares. 

Rule. —  To  the  market  value  of  one  share  ada 
the  brokerage  (if  any),  and  divide  the  sum  to  be 
invested  by  the  amount  thus  obtained. 


84        ORTON  &  Sadler's  calculator. 

Example. — How  raany  shares  of  Baltimore 
&  Ohio  Railroad  stock  can  be  purchased  for 
$12,750?  Market  value  1271,  brokerage  i%. 
Ans.  100  shares. 

p:^ocess. 

127%  (Market  value)  -fV^  (Brokerage)  =  V2.Vyi  Ccst  of  1  share. 
ei2,750  ^  127}^  =  100  Shares. 


To  ascertain  amount  of  Investment. 

Rule. — Divide  the  stated  income  by  the  income 
on  one  share  (which  will  give  the  number  of 
shares  required).  Multiply  the  number  of  shares 
by  the  cost  per  share,  and  the  product  will  be  the 
required  investment. 

Example. — How  much  capital  must Ije  invest- 
ed in  New  York  Central  Railroad  stocks  @  110, 
which  pay  semi-annual  dividends  of  6%,  to  realize 
^n  income  of  $900  per  annum  ?     Ans.  $8250. 
PROCESS. 

SOOO  (Desired  income)  -T-$r2  (Income  on  1  share)  =75  No.  of  shares. 
$110  (Coat  o!  1  share)  X  75  (Number  of  shares)  —  ■$>S2o{)  luvestmeut. 


To  ascertain  the  Rate  %  of  iiihome  realized 
from  investments. 

Rule. — Divide  the  annual  dividend  or  income 
on  one  share  by  the  cost  per  share. 

Example. — If  Railroad  shares  paying  annual 
dividends  of  10%  command  a  premium  of  25% 
— what  per  cent,  of  income  will  be  realized  from 
investing  in  said  shares?     Ans,  8%. 


INVESTMENTS.  85 

PROCESS. 
^0  (Income  from  I  share)  ~-  $125  (Ccst  of  1  share^sr?^. 


To  ascertain  at  what  price  stocks  must  be  bought 
to  produce  a  certain  Income. 

Rule. — Divide  the  dividend  or  income  on  one 
share  by  the  desired  rate  of  interest  or  income. 

Example  I. — What  amount  of  premium  must 
stocks  bring,  paying  annual  dividends  of  12%, 
to  net  9%  income  to  the  investor?  Arts,  33  J  % 
Premiunf. 

PROCESS. 

$12  (Income  from  1  8haro)-^9^  (Keq.  \nt)=$l23%  Value  of  1  share. 
1333^—100  Par  value=33>^  Premium. 

Example  II. — At  what  price  must  stock  pro- 
ducing annual  dividends  of  6%  be  bought  so  as 
to  net  the  investor  9%.    Ans.  33i%  discount. 

PROCESS. 

S6  (Income  from  1  8harc)-H9«<  CRcq.  int.)=?66^.<  Value  of  1  share. 
jlOO  (Par  value  of  1  share) — CG%  (Market  value)=33)/;^  DifecouuL 


TEST   EXAMPLE. 


At  what  price  must  stock  of  the  par  value  of 
$50  per  share,  which  pays  annual  dividends  of 
$3  per  share,  be  bought  to  produce  an  income 
of7J%?    Ans.$iO. 
8 


S6        ORTON  &,  Sadler's  calculator. 

TABLE  FOR  INVESTORS. 

The  following  Table  shows  the  rate  per  cent,  of  Annual 

Income  from  Bonds  bearing  5,  6,  or  7  per  cent. 

interest  J  and  costing  from  50  to  125. 


Purchase 
Price. 

5% 

6% 

n 

Purchase 
Price. 

5% 

6% 

7% 

50 

10.00 

12.00 

14.00 

88 

6.68 

6,81 

7.94 

61 

9.80 

11.76 

13.72 

89 

6.61 

6.74 

7.86 

52 

9.61 

11.53 

13.46 

90 

5.65 

6.66 

7.77 

53 

9.43 

11.32 

13.20 

91 

5.49 

6.69 

7.69 

54 

9.25 

11.11 

12.96 

92 

6.43 

6.52 

7.60 

55 

9.00 

10.90 

12.72 

93 

6.37 

6.4.5 

7.52 

56 

8.92 

10.70 

12  60 

94 

5.31 

6.38 

T.44 

57 

8.77 

10.52 

12.27 

95 

6.26 

6.31 

7.36 

68 

8.62 

10.34 

12.06 

96 

5.20 

6.25 

7.29 

69 

8.47 

10.16 

11.86 

97 

S.15 

6.18 

7.21 

60 

8.33 

10.00 

11.66 

98 

6.10 

6.12 

7.14 

61 

8.19 

9.83 

11.47 

99 

5.05 

6.06 

7.07 

62 

8.06 

9.67 

11.29 

ICO 

6.C0 

6.00 

7.C0 

63 

7.93 

9.52 

11.11 

101 

4.95 

5.94 

6.93 

64 

7.81 

9.37 

10.93 

102 

4.90 

5.88 

6.86 

65 

7.69 

9.23 

10.76 

103 

4.85 

5.82 

6.79 

66 

7.57 

9.09 

10.60 

104 

4.80 

6.76 

6.72 

67 

7.46 

8.95 

10.44 

105 

4.76 

5.71 

6.66 

6« 

7.35 

8.82 

10.29 

106 

4.71 

6C6 

6.60 

69 

7.24 

8.69 

10.14 

107 

4.67 

6.60 

0.64 

70 

7.14 

8.57 

10.00 

108 

4.02 

6.55 

6.48 

71 

7.04 

8.45 

9.85 

109 

4.68 

5.50 

6.42 

72 

6.94 

8.33 

9.72 

110 

4.54 

5.45 

6.36 

73 

6.84 

8.21 

9.58 

111 

4.60 

6.40 

6.30 

74 

6.75 

8.10 

9.45 

112 

4.46 

5.35 

6.26 

75 

6.60 

8.00 

9.33 

113 

4.42 

6.30 

6.19 

76 

6.67 

7.89 

9.21 

114 

4.38 

6.26 

6.14 

77 

6.49 

7.79 

9.(10 

115 

4.35 

5.21 

6.08 

78 

6.41 

7.69 

8.97 

116 

4.31 

5.17 

6.03 

79 

6.32 

7.59 

8.86 

117 

4.27 

6.12 

5.98 

80 

6.25 

7.cO 

8.75 

118 

4.23 

6.08 

5.93 

n 

0.17 

7.40 

8.64 

119 

4.20 

6.04 

6.88 

82 

6.09 

7.31 

8.53 

120 

4.16 

5.C0 

5.^3 

83 

6.02 

7.22 

8.43 

121 

4.13 

4.95 

5.78 

84 

5.95 

7.14 

8.33 

122 

4.09 

4.91 

5  73 

85 

6.88 

7.05 

8.23 

123 

4.06 

4.87 

5.69 

86 

5.81 

6.97 

813 

124 

4.03 

4.83 

5.65 

87 

6.74 

6.89 

8.04 

125 

4.00 

4.80 

6.60 

Definition  of  Terms 

Interest  is  premium  paid  for  the  use  of  money, 
goods,  or  property. 

It  is  computed  by  percentage  —  a  certain  per 
cent,  on  the  money  being  paid  for  its  use  for  a 
stated  time.  The  money  on  which  interest  is  paid 
is  called  the  principal. 

The  per  cent,  paid  is  called  the  rate  ;  the  prin- 
cipal and  interest  added   together  is  called   the 

AMOUNT. 

When  a  rate  per  cent,  is  stated,  without  the 
mention  of  any  term  of  time,  the  time  is  under- 
stood to  be  I  year. 

The  first  important  step  in  the  calculation  of 
simple  interest  is  the  arranging  of  the  time  for 
which  it  is  computed.  The  student  must  study  the 
87 


88        ORTON  &  Sadler's  calculator. 

following  Propositions  carefully,  if  he  would  be 
expert  in  this  important  and  useful  branch  of  buB- 
iness  calculations : 

PROPOSITION  1. 
£f  the  time  consists  of  years^  multiply  the  principal 
by  the  rate  per  cent,,  and  that  product  by  the 
number  of  years. 

Example  1. — Find  the  interest  of  $75  for  4 
years  at  6  per  cent. 
Operation. 

$75  The  decimal  for  6  per  cent,  is 

.06  06.     There  being  two  places  of 

decimals  in  the  multiplier,  we 

4.50  point  off  two  in  the  product. 

4 


$18.00  Ans. 

PROPOSITION  2. 
If  the  time  consists  of  years  and  mx)nt}iSy  reduce  the 

time  to  months^  and  multiply  the  principal  hy  the 

rate  per  cent,  and  number  of  months  together^  and 

divide  the  result  by  \2. 

Note. — The  work  can  always  be  abbreviated  at 
4,  6,  8,  9,  12,  and  15  per  cent.,  by  canceling  the 
per  cent.,  or  time,  or  principal,  with  the  common 
diTiuor  12. 


INTEREST. 


89 


Example  :i..-— Find  the  interest  of  J240  for  2 
/ears  and  7  months  at  8  per  cent. 


First  method. 
Principal, 
Per  cent., 

In.  for  lyr., 
2yrs.+7mo3., 


$240 
.08 

19.20 
31mos. 


12)595.20 


Second  method : 
by  cancellation. 
^^0—20 
8  rate. 

31  time. 


It 


49.60  Am, 


$49.60  Am. 

The  operation  by  canceling  is  much  more  brief. 
We  simply  place  the  principal,  rate,  and  time,  an 
the  right  of  the  line,  and  12  on  the  left;  then  we 
cancel  12  in  240,  and  the  quotient  20  multiplied 
with  8  and  31  gives  the  interest  at  once. 

Note. — After  12  is  canceled  the  product  of  the 
remaining  numbers  is  always  the  interest. 

PROPOSITION  8. 

If  the  time  consists  of  years^  months^  and  days,  re- 
duce the  years  to  months,  add  in  the  given  months^ 
and  place  one-third  of  the  days  to  the  right  of 
this  number,  which  we  multiply  by  the  principal 
and  rate  per  cent.,  and  divide  by  12,  as  before, 
or  cancel  and  divide  by  12  before  multiplying. 
Example  3. — Find  the  interest  of  $231  for  1 

^eur,  1  month,  and  6  days,  at  5  per  cent. 


90 


OETON  &   SADLER'S  CALCULATOR. 


First  method. 

Second  method : 

Principal, 

«231 

by  cancellation. 

Per  cent., 

.05 

231          pria 

i$ 

5            rate. 

In,  for  lyr., 

11.55 

m-n 

lyr.+  lmo.+6da. 

,    13.2mo, 

- 

ei2.705  An$. 

12)152.460 

$12,705  Am. 

By  the  second  method  we  cancel  12  in  132,  and 
multiply  the  quotient  11  by  5  and  231. 

Note. — When  the  principal  is  $,  and  the  time  is 
in  years  or  months,  the  interest  is  in  cents ;  if  the 
time  is  in  years,  months,  and  days,  the  interest  ife 
m  mills,  unless  the  days  are  less  than  3,  in  which 
case  it  would  be  in  cents,  as  before. 

Note. — The  reason  we  divide  the  days  by  3  is 
because  we  calculate  30  days  for  a  month,  and  di- 
viding by  3  reduces  the  days  to  the  tenth  of  months. 

Note. — The  three  preceding  propositions  will 
work  any  note  in  interest  for  any  time  and  at  any 
given  rate  per  cent. 

How  to  Avoid  Fractions  in  Interest, 

PROPOSITION  4. 

jy,  when  the  time  coiisists  of  years^  months^  and 

days,  are  not  divisible  by  3,  you  can  divide  the  days 

by  3,  and  annex  the  mixed  number  as  in  F'^'tsyosition 


INTEREST. 


91 


3,  on/  you  wish  to  avoid  fractions^  you  can  reduce 
the  time  to  interest  days,  and  multiply  the  principal^ 
rate  and  days  together ,  and  divide  the  result  by  36 
or  its  factors,  4X  9. 

Note. — la  this  case  as  in  the  preceding,  the 
work  can  almost  always  be  contracted  by  dividing 
the  rate  or  time  or  principal  with  the  divisor  36. 

Note. — We  use  the  divisor  36,  because  we  cal- 
culate 360  interest  days  to  the  year.  We  discard 
the  0,  because  it  avails  nothing  to  multiply  or  di- 
vide by. 

Example  4.— -Find  the  interest  of  $210  for  1 
year,  4  months,  and  8  days,  at  9  per  cent. 

Year.        Months.        Days. 


1             4 

8=16.2J  months  or  48?  days 

Operation 

Operation 

By  Prop.  3. 

By  Prop.  4. 

$210 

$210 

.9 

9 

18.90 
16.2S 


12)307440 


$25,620  Ans, 


18.90 

488 


36)922320 


$25,620  Am 


We  will  now  work  the  example  by  cancellation 
to  show  its  brevity. 


92        ORTON  &  Sadler's  calculator. 

Operation  hy  Gancellatum. 

Time  488  days. 

210 
4-J0       0 

m$  122 

122 
210 


$25,620 

Now  cancel  9  in  36  goes  4  times,  then  4  into  488 
goes  122.  Now  multiply  remaining  numbers  to- 
gether, thus,  210x122  and  we  have  the  interest  at 
once. 

When  the  days  are  not  divisible  by  3  we  reduce 
the  whole  time  to  days ;  then  we  placo  the  princi- 
pal rate  and  time  on  the  right  of  the  line.  Now, 
because  the  time  is  in  days,  we  place  36,  on  the 
left  of  the  line  for  a  divisor.  {1/  the  time  wcu 
months  we  would  place  12  on  the  left.) 

Note. — A  very  short  method  of  reducing  time 
to  interest  days  is  to  multiply  the  years  by  36 ; 
add  in  3  times  the  number  of  months  and  the  tons* 
ftgure  of  the  days,  and  annex  the  unit  figure ;  but 
If  the  days  are  less  than  10  simply  annex  them. 

EXAMPLE  1. — Reduce  1  year,  2  months,  and  6 
days,  to  days. 

Tears.        Months.  Days. 

36X1+3X2=42  annex  6=426  Am. 


SIMPLE  INTEREST  BT  CANCELLATION.  93 

Example  2. — Beduce  2  years,  3  months    and 
17  days,  to  interest  days. 

Tears.    M'ths.  Days.  Days. 

36x2+3x3+1=82.  annex  7=827  days  Aua. 
Note. — The  student  should  commit  to  meiiiory 
the  multiplication  of  the  number  36  up  as  far  as 
9  times   36,  and  then  he  can  reduce  almost  in- 
stantly years,  months,  and  days,  to  days. 


SIMPLE  INTEREST  BY  CANCELLATION. 

Rule. — Place  the  principal^  time^  and  rate  per 
cent  on  the  right  hand  side  of  the  line.  If  the  time 
consists  of  years  and  months^  reduce  them  to  months j 
and  place  12  (the  number  of  months  in  a  i/ear)  on 
the  left  hand  side  cf  the  line.  Should  the  tims  con- 
mt  of  months  and  days^  reduce  them  to  days  or  deci- 
mal parts  of  a  m^onth.  If  reduced  to  days,  place 
36  on  the  left.  If  to  decimals  parts  of  a  month, 
place  12  only  as  before. 

Point  off  two  decimal  plaices  when  the  time  is  in 
months,  and  tnree  decimal  places  when  the  time  is 
in  days. 

Note.  If  the  principal  contains  cents,  point  oflF 
four  decimal  places  when  the  time  is  in  monthn 
and  five  decimal  places  when  the  time  is  in  day^. 


94   ORTON  &  Sadler's  calculator. 

Note. —  Wt  'place  36  on  the  left  hecause  there  ^4 
300  interest  days  in  a  year,  (^Ctistom  has  made  tht& 
lawful.') 

Example  1. — What  is  the  interest  on  $60  foi 
117  days  at  6  per  cent? 


Operation. 

Here  117X0 

00 

Both  sixes    on    the 

must  be  the  $0 

0 

right  cancels  36  on 

answer. 

117 

the    left,    and    we 
have    nothinir    left 

$1,170  Am.     to  divide  by. 
In  this  case  we  point  oflf  three  decimal  places  be- 
cause the  time  is  in  days.    If  the  time  had  been  111 
months,  we  would  have  pointed  oflf  but  two  deci 
mal  places. 

Example  2. — What  is  the  interest  of  S96.50 
for  90  days  at  6  per  cent? 
Operation. 

96.50  9650 

0— $0     00 — 15  15 

0  

1.44.750  An$, 
Now  cancel  6  in  36   and  the  quotient  6   into 

90,  and  we  have  no  divisor  left.     Hence  15x96.50 

must  be  the  answer. 

Note — As  there  are  cents  in  the  principal,  we 

point  off  five  decimals  ;  three  for  days  and  two  foi 

oents      Pay  no  attention  to  the  decimal  point  trntil 

ihe  close  of  the  operation. 


SIMPLE   INTEREST   BT   OANOELLAnON.         95 


Example  3. — What   is   the   interest   of  $480 

for  361  days  at  6  per  cent? 

^g0— 80                      361 

0— J0 

361                                   80 

828.880  Am. 

Now  cancel  6  in  36  and  the  quotient  6  into  480, 

and  we  have  no  divisor  left.     Hence  80x361  must 

be  the  answer. 

Example  4.— What  is  the  interest  of  $720  for 

9  months  at  7  per  cent? 

tn—^^                60 

n 

9                                         9 

7                                   

540 

7 


$37.80  Arts, 
Now  cancel  12  in  720  there  is  nothing  left  to 
divide  by.     Hence  60x9x7  must  be  the  answer. 

N.  B.  When  interest  is  required  on  any  sum  for 
days  only,  it  is  a  universal  custom  to  consider  30 
days  a  month,  and  12  months  a  year  ;  and,  as  the 
unit  of  time  is  a  year,  the  interest  of  any  sum  for 
one  day  is  j^^,  what  it  would  be  for  a  year.  For 
2  days,  3§o,  etc.;  hence  if  we  multiply  by  the 
days,  we  must  divide  by  360,  or  divide  by  36  and 
save  labor.  The  old  form  of  this  method  was  to 
place  360,  or  12  and  30,  on  the  left  of  the  line, 
but  using  36  is  much  shorter. 


96        OBTOK  dc  Sadler's  calculator. 

WHEN    THE   DAYS   ARE  NOT  DIVISIBLE    BT  THEEBi 

Note. — When  the  time  consists  of  months  and 
days,  and  the  days  are  not  divisible  by  three,  re- 
duce  the  time  to  dayi. 

Example  5  —What  is  the  interest  of  *960  fof 
1 1  months  and  20  days  at  6  per  cent? 

Months.    DajB. 

Operation.  11     20=350  days. 

000—160  350 

^  —36     350  160 

6  

$56,000 

Now  cancel  6  in  36  and  the  quotient  6  into 
960,  and  we  have  no  divisor  left.  Hence  160X 
350  must  be  the  answer. 

Example  6. — What  is  the  interest  of  $173  for 
8  months  and  16  days  at  9  per  cent? 

Months.    DayB. 

Operation.  8     16=256  dayg, 

173  173 

4r-H    0  64 

??0— 64  

$11,072  Aru, 
Now  cancel  9  in  36  and  the  quotient  4  into  256, 
and  we  have  no  divisor  left.    Hence  64X173  must 
be  the  answer. 

N.  B.  Let  the  puj  il  remember  that  this  is  a  gen- 
eral and  universal  method,  equally  applicable  to 
any  per  cent,  or  any  required  time,  and  all  other 
rules  must  be  reconcilable  to  it ;  and,  in  &ct,  all 
other  rules  are  but  modifications  of  this. 


SIMPLE  INTEREST  BY  CANCELLATION.    97 

Example  7. — What  is  the  interest  on  $1080 
Tor  7  months  and  11  days  at  7  per  cent? 

UoQtks.    Days. 

7         11=221  days. 

Operation. 

10$0— 30  221 

10       221  30 

7  

6630 
7 


$46,410  Ans. 
Now  cancel  36  in  1080  and  we  have  no  divisor 
left,  hence  30X221X7  must  be  the  answer. 

WITH  more  difficult  TIME  AND  RATE  PER  CENT. 

Example  8. — What  is  the  interest  of  $160  for 
19  months  and  23  days  at  4^  per  cent? 

Months.    Dajb. 

19        23  =-593  days. 
Operation. 

160—20  593 

t^H    593  20 

^  

$11,860  Ans. 
Now  cancel  4|  in  36  and  the  quotient  8  into  160 
we  have  no  divisor  left,  hence  20x593  must  be 
the  interest. 

when    the   DATS   ARE   DIVISIBLE    BY   THREE. 
EuLE. — Place  <me-thxrd  of  the  days  to  the  ri'ghi 
^/  the  months,  and  place  12  on  the  left  of  the  line, 
9 


:JS 


ORTON   &   SADLER  8    CALCULATOR. 


Example  11. — What  is  the  interest  of  $350  for 
3  years  7  months  and  6  days  at  10  per  cent? 

Tears.    Monthi.    Days. 

3  7  6=43.2  months. 


Operation. 
350 


It 


10 


-36 


350 
36 


12600 
10 


$126,000  Aru. 

Now  cancel  12  in  432  and  we  have  no  divisor 
left.     Hence  350x36x10  equals  the  interest. 

"Example  12. — What  is  the  interest  of  8241  for 
I     months  and  9  days  at  8  per  cent? 

Months.    Days. 

13        9=13.  3  months. 


Operation. 
241 


3-1? 


13.3 

$-2 


241 
133 


32053 
2 


3)64106 


$21.368§^rwi. 
In  this  example  I  canceled  8  and  12  by  4,  and 
then  multiplied  all  on  the  right  of  the  line  and  di- 


SIMPLE  INTEEEST  BY  CANCELLATION.    99 

rided  by  3.  If  I  could  have  divided  by  3  before 
multiplying  I  would  have  saved  labor,  but  when  the 
Dumbers  are  prime  the  whole  work  must  be  liter- 
ally done. 

Closinq  Remarks. — We  have  now  fully  ex 
plained  the  canceling  system  of  computing  inter- 
est. Any  and  every  problem  can  be  stated  by  this 
method,  and  the  beauty  and  simplicity  of  the 
system  ranks  it  high  among  the  most  important 
abbreviations  ever  discovered  by  man.  As  we  have 
before  remarked,  at  6,  4,  8,  9,  12,  15,  and  4J  per 
cents.,  every  problem  in  interest  can  be  canceled, 
besides  a  great  many  can  be  abbreviated  at  5,  7,  and 
other  per  cents.;  and  after  the  problem  has  been 
stated  and  we  find  that  we  can  not  cancel,  what 
have  we  done?  We  have  simply  stated  the  prob- 
lem in  its  simplest  and  easiest  form  for  working  it 
by  any  other  method.  Hence  we  have  a  decided 
advantage  of  all  notes  that  will  cancel,  and  if  we 
can  not  cancel  we  have  stated  the  problem  in  ita 
correct  and  proper  form  for  going  through  the 
whole  work;  but  it  is  only  when  the  principal, 
time,  and  rate  per  cent,  are  all  prime,  that  the 
WHOLE  work  must  be  literally  done.  At  6  per 
cent,  we  can  cancel  through,  and  6  is  the  rate  mo$* 
commonly  used 


fc.  t^»  ^•«!^   *^r* 

SHORT    PRACTICAL    RULES, 

DEDUCED  FROM  THE  CANCELING  SYSTEM, 

For  calculating  interest  at  6  per  cent.,  either  for 
monthSj  or  months  and  days. 

To  find  the  interest  for  months  at  6  per  cent. 

Rule. — Multiply  the  principal  hy  half  the  num- 
her  of  months  J  expressed  decimally  as  a  per  cent., 
that  is,  for  12  months,  multiply  hy  .06 ;  for  8  months^ 
multiply  hy  .04. 

Note  1. — It  is  obvious  that  if  the  rate  per  cent 
were  12,  it  would  be  1  per  cent,  a  month  ;  if,  there- 
fore, it  be  6  per  cent.,  it  will  be  a  half  per  cent,  a 
month ;  that  is,  half  the  months  will  be  the  per 
cent. 

Note  £. — If  any  other  per  cent,  is  wanted  you 
can  proceed  as  above,  and  then  multiply  by  the 
i^iven  rate  per  cent,  and  divide  by  6,  and  the  quo- 
tient ii  the  interest. 

1.  Wh&t  is  the  interest  of  $368  for  8  months? 

S368 
.04=half  the  months. 


»14.72=^ns. 
Note  3, — When  the  months  are  not  even;  that  is, 
will  not  divide  by  2,  multiply  one-half  the  principal 

100 


SHORT  PRACTICAL  RULES.        101 

by  the  whok  number  of  months,  expressed  deoi 
mally. 

To  find  the  interest  of  any  sum  at  6  per  cent, 
per  annum  for  any  number  of  months  and  days. 

KuLE. — Divide  the  days  hy  3  and  place  the  quo- 
tient to  the  right  of  the  months;  one-half  of  the  num- 
ber thiis  formed  multiplied  hy  the  principal^  or  one- 
half  of  the  principal  multiplied  hy  this  number,  will 
give  the  interest — pointing  off  three  decimal  places 
when  the  principal  is  $, 

2.  What  is  the  interest  of  $76  for  1  year-  6 
months,  and  12  days,  at  6  per  cent? 

Tears.  Months.  Days. 

1        6        12=18.4  months— half  9  2. 
$76  Or,  184 

9.2  38=half  priu. 


$6,992  Ans.  $6,992  Ans, 

Note. — Dividing  the  days  by  3  reduces  them  to 
the  tenth  of  months. 

To  find  the  the  interest  of  any  sum  at  6  per 
cent,  per  annum  for  any  number  of  days. 

Rule. — Divide  the  principal  hy  6  and  multiply 
the  quotient  by  the  number  of  days ;  or  divide  the 
days  hy  6  and  multiply  the  quotient  hy  the  principal^ 
pointing  off  three  decimal  places  when  the  principal 

M$ 

Note. — Always  divide  6  into  the  number  that 


102   ORTON  &  Sadler's  calculator. 

will  divide  without  a  remainder;  if  neither  one 
will  divide,  multiply  the  principal  and  days  to- 
gether and  divide  the  result  by  6. 

3.  What  is  the  interest  of  8240  for  18  days  at 
6  per  cent  ? 

18-^6=3  240-5-6=40 

8240  Or,  840=i  of  prin. 

3=J  of  the  days.  18 

80.720  Am.  80.720  Am. 

4.  What  is  the  interest  of  81800  for  72  days  at 
6  per  cent. 

81800  Or,  8300=J  of  prin. 

12=^  of  the  days.  72 

821.600  Am.  821.600  Am. 

Useful  Suggestions  to  the  Accountant  in  Computing 
Interest  at  6  per  cent. 

If  the  principal  is  divisible  by  6,  always  reduce 
the  time  to  dags;  then  multiply  the  number  of 
days  by  one -sixth  of  the  principal. 

EXAMPLE. 

5.  Find  the  interest  of  8240  for  1  year,  5  months 
and  17  days,  at  6  per  cent. 

6)240  lyr,  5mos.,  17da.=527  days, 

Multiplied  by     40 

I  of  prin.=:40  

821.080  An$. 


SHORT  PRACTICAL  RULES.        103 

If  the  days  are  only  divisible  by  3,  multiply 
one-third  of  the  principal  by  one-half  of  the  days 

6.  What  is  the  interest  of  $210  for  80  days  at  6 
per  cent.  ? 

$70= J  of  the  principal. 
40=^  of  the  days. 

$2,800  Ans, 

When  the  Rate  of  Interest  is  4  per  cent 
Rule. — Multiply  the  principal  hy  one-third  the 
number  of  months,  or  hy  one-ninth  the  number  of 
days,  and  the  product  is  the  interest. 

Note. — This  principle  is  also  deduced  from  the 
canceling  method  of  computing  interest ;  the  stu- 
dent can  readily  see  that  4  is  J  of  12  and  I  of  36, 

When  the  Mate  of  Interest  is  9  per  cent. 
Rule. — Multiply  the  principal  by  three-fourths 
the  number  of  months,  or  one-fourth  the  number  of 
days^  or  vice  versa. 


BANKS  AND  BANKING. 


A  Bank  is  an  institution  established  under  legal  charter 
for  the  purpose  of  trafficking  in  money. 

They  issue  notes  payable  to  bearer  on  demand,  which 
circulate  as  money,  receive  money  on  deposit,  and  make 
loans. 

Money  Deposit  in  Banks  is  generally  subject  to  the 
order  of  the  depositor  by  check. 

A  Bank  Check  is  the  written  order  of  the  depositor 
for  the  payment  of  money. 

Notes. — In  discounting  business-paper  it  is  customary  for  banks  to 
ciilculate  or  take  off  the  interest  for  the  actual  number  of  days  from  the 
day  of  discount  to  date  of  maturiti/,  ])oth  days  inclusive. — See  Bank 
Discount,  page  76. 

"  How  to  Tmnsact  Business  with  Banks," — See  page  281. 

The  endorser  of  a  note  incurs  all  (he  obligations  of  such  endorse- 
ment, even  though  he  may  be  ignorant  of  the  law  at  the  time. 

Many  a  mar  has  been  reduced  from  affluence  to  poverty  by  merely 
writing  his  name  on  the  back  of  a  note,  '^Just  to  accommodate  a 
friend.'* 

104 


:coHPu!i|[c!sTfREST.: 


AT  6  PER  CENT.  FOR  ANY  NUMBER  OF  DATS. 


BULE. — Draw  a  perpendicular  linCy  cutting  off 
ih^  two  right  hand  figures  of  the  $,  and  you  havt  the 
Vfterest  of  the  sum  for  60  days  at  6  per  cent. 

Note. — The  figures  on  the  left  of  the  line  are  $, 
and  those  on  the  right  are  decimals  of  $. 

Example  1. — What  is  the  interest  of  $423 
60  days  at  6  per  cent.  ? 

$423=the  principal. 

H  I  23  cts.=interest  for  60  days. 

Note. — When  the  time  is  more  or  less  than  60 
days,  first  get  the  interest  for  60  days,  and  from 
that  to  the  time  required. 

Example  2. — Wh%t  is  the  interest  of  $124  for 
15  days  at  6  per  cent.? 

Days.  Days. 

15z=J  of  60 
$124=:principal. 
4)1  I  24  ct8.=interest  for  60  days.  i 

I  31  ots=interest  for  15  days. 
105 


106   ORTON  &  Sadler's  calculator. 

Example  3.— What  is  the  interest  of  8123.40  foi 
90  days  at  6  per  cent.? 

Dayb.  Days.  DayB. 

90—60+30 
$123.40=principal. 
2)1  I  2340r=interest  for  60  days. 
I  6170=interest  for  30  days. 


Ans.   $1  I  851=interest  for  90  days. 

Example  4. — What  is  the  interest  of  J324  for 
75  days  at  6  per  cent.? 

Days.  Daya.  Days. 

$324=principal.  75=60+ 15 


4)3 


24  cts.  interest  for  60  days. 
81  cts.  interest  for  15  days. 


Ans,  $4  I  05  cts.  interest  for  75  days. 

Kemarks. — This  system  of  Computing  Interest 
is  very  easy  and  simple,  especially  when  the  days 
are  aliquot  parts  of  60,  and  one  simple  division 
will  suffice.  It  is  used  extensively  by  a  large  ma- 
jority of  our  most  prominent  bankers ;  and,  indeed, 
is  taught  by  most  all  Commercial  Colleges  as  the 
shortest  system  of  computing  interest. 

Method  of  Calculating  at  Different  Per  Cents. 
This  principle  is  not  confined  alone  to  6  percent. 
ds  many  suppose  who  teach  and  use  it.  It  is  their 
custom  first  to  find  the  interest  at  6  per  cent.,  and 
from  that  to  other  per  cents.  But  it  is  equally  ap 
plicable  foi  all  per  cents.,  from  1  to  15  inclusive. 


bankers'  method  of  computing  interest.  107 

The  following  table  shows  the  different  per  cents., 
with  the  time  that  a  given  number  of  $  will 
amount  to  the  same  number  of  cents  when  placed 
at  interest. 

Rule. — Draw  a  perpendicular  line^  cutting  oS 
the  two  right  hand  figures  of  $,  and  you  have  the 
interest  at  the  following  per  cents. 

Interest  at  4  per  cent,  for  90  days. 

Interest  at  5  per  cent,  for  72  days. 

Interest  at  6  per  cent,  for  60  days. 

Interest  at  7  per  cent,  for  52  days. 

Interest  at  8  per  cent,  for  45  days. 

Interest  at  9  per  cent,  for  40  days. 

Interest  at  10  per  cent,  for  36  days. 

Interest  at  12  per  cent,  for  30  dajs. 

Interest  at  7-30  per  cent,  for  50  days. 

Interest  at  5-20  per  cent,  for  70  days. 

Interest  at  10-40  per  cent,  for  35  days. 

Interest  at  7^  per  cent,  for  48  days. 

Interest  at  4^  per  cent,  for  80  days. 
Note. — The  figures  on  the  left  of  the  perpen- 
dicular line  are  dollars,  and  on  the  right  decimals 
of  $.     If  the  $  are  less  than  10  prefix  a  0. 

Example  1. — What  is  the  interest  of  ?120  for 
15  days  at  4  per  cent? 

Days.  Daya. 

$120=principal.  15=rJ  of  90. 

6)1  I  20  cts.-  ini  for  90  days. 
20  cts.^int.  for  15  days. 


108      ORTON  &  Sadler's  calculator. 

Example  2. — What  is  the  interest  of  $132  for 
13  days  at  7  per  cent.  ? 

Days.  Days. 

$132=principal.  13=J  of  52. 

4)1  I  32  cts.rrrint.  for  52  days. 
I  33  cts.=int.  for  13  days. 
Example  3. — What  is  the  interest  of  $520  for 
9  days  at  8  per  cent.  ? 

Days.  Days. 

$520=--principal.  9=^  of  45. 

5)5  I  20  cts.=int.  for  45  days. 
$1  I  04  cts.=int.  for  9  days. 

Example  4. — What  is  the  interest  of  $462  for 
for  64  days  at  7J  per  cent.  ? 

Days.  Days.  Days. 

$462=::principal.  64=48+16. 

3)4  I  62  cts.  =:int.  for  48  days. 
1  I  54  cts.  =:int.  for  16  days. 


$6  I  16  cts.=:=int.  for  64  days. 

Eemark. — We  have  now  illustrated  several  ex- 
amples by  the  dififerent  per  cents. ;  and  if  the  stu- 
dent will  study  carefully  the  solution  to  the  above 
examples,  he  will  in  a  short  time  be  very  rapid  in 
this  mode  of  computing  interest. 

Note. — The  preceding  mode  of  computing  in- 
terest is  derived  and  deduced  from  the  canceling 
system ;  as  the  ingenious  student  will  readily  see. 
It  is  a  short  and  easy  way  of  finding  interest  for 
days  when  the  days  are  even  or  aliquot  parts ;  but 
when  they  are  not  multiples,  and  three  or  four  di- 


BANKERS'  METHOD  OP  COMPUTING  INTEREST.    109 

visions  are  ncessary,  the  canceling  system  is  muoh 
more  simple  and  easy.  We  will  here  illustrate  an 
example  to  show  the  difference :  Required  the  in- 
terest of  $420  for  49  days  at  6  per  cent. 

Bankers*  method.  Canceling  meth. 

20  cts.=int.  for  60  days. 


2)4 

2)2 
5)1 
3) 


0— ?0 


10  cts.=int.  for  30  days. 
05  cts.=int.  for  15  days. 

21  cts.=int.  for  3  days.  

7  cts.=:int.  for  1  day.  $3,430  Am. 


0 
49 
70 


$3  I  43  cts.=int.  for  49  days. 

The  canceling  method  is  much  more  brief;  we 
simply  cancel  6  in  36,  and  the  quotient  6  into  420  ; 
there  is  no  divisor  left;  hence  70X49  gives  the  in- 
terest at  once. 

If  the  time  had  been  15  or  20  days,  the  Bankers' 
Method  would  have  been  equally  as  short,  because 
15  and  20  are  aliquot  parts  of  60.  The  superiority 
the  canceling  system  has  above  all  others  is  this :  it 
takes  advantage  of  the  principal  as  well  as  the  time. 

For  the  benefit  of  the  student,  and  for  the  con- 
venience of  business  men,  we  will  investigate  this 
system  to  its  full  extent  and  explain  how  to  take 
advantage  of  the  principal  when  no  advantage  can 
be  taken  of  the  days.  This  is  one  of  the  most  im- 
portant characteristics  of  interest,  and  very  often 
saves  much  labor.  It  should  he  used  when  the  d<u/$ 
are  not  even  or  aliquot  parts, 
10 


110      ORTON  &  Sadler's  calculator. 

The  following  table  shows  the  different  sams  of 
money  (at  the  different  per  cents.)  that  bear  1  cent 
interest  a  day ;  hence  the  time  in  days  is  always 
the  interest  in  cents ;  therefore,  to  find  the  interest 
on  any  of  the  following  notes  at  the  per  cent,  at- 
tached to  it  in  the  table,  we  have  the  following 
rule: 

Rule. — Draw  a  perpendicular  line^  cutting  off 
the  two  right  hand  figures  of  the  days  for  cents,  and 
you  have  the  interest  for  the  given  time. 

Interest  of  $90  at  4  per  cent,  for  1  day  is  1  cent 

Interest  of  S72  at  5  per  cent,  for  1  day  is  1  cent. 

Interest  of  S60  at  6  per  cent,  for  1  day  is  1  cent. 

Interest  of  $52  at  7  per  cent,  for  1  day  is  1  cent. 

Interest  of  $45  at  8  per  cent,  for  1  day  is  1  cent. 

Interest  of  $40  at  9  per  cent,  for  1  day  is  1  cent. 

Interest  of  $36  at  10  per  cent,  for  1  day  is  1  cent. 

Interest  of  $30  at  12  per  cent,  for  1  day  is  1  cent. 

Interest  of  $50  at  7.30  per  ct.  for  1  day  is  1  ct. 

Interest  of  $70  at  5.20  per  ct.  for  I  day  is  1  ct. 

Interest  of  $35  at  10.40  per  ct.  for  1  day  is  1  ct. 

Interest  of  $48  at  7J  per.  cent,  for  1  day  is  1  sent. 

Interest  of  $80  at  4^  per  cent,  for  1  day  is  1  cent. 

Interest  of  $24  at  15  per  ct.  for  1  day  is  1  cent. 

Note. — The  7.30  Government  Bonds  are  calcu- 
lated on  the  base  of  365  days  to  the  year,  and  the 
5.20'a  and  10.40's  on  the  base  of  364  days  to  the  year. 


BANKERS    METHOD  OF  COMPUTING  INTEREST.   Ill 

Note. — This  table  should  be  committed  to  mem- 
01  y,  as  it  is  very  useful  when  the  days  are  not  even 
or  aliquot  parts.  If  the  days  are  less  than  10  pre- 
fix a  0  before  drawing  the  line. 

ExA.MPLE  1. — Required  the  interebt  of  $60  far 
117  days  at  6  per  cent. 

117=  the  days.  Here  we  cut  off  the  two 

?1  I  17  cts.  Ans.        right  hand  figures  for  cents. 

The  student  should  bear  in  mind  that  the  inter- 
est on  ?60  for  117  days  is  just  the  same  as  the 
interest  on  $117  for  60  days. 

By  looking  at  the  table  we  see  that  the  interest 
for  $60  at  6  per  cent,  is  1  cent  a  day ;  hence  the 
time  in  days  is  the  answer  in  cents.  If  this  note 
was  $120,  instead  of  $60,  we  would  first  find  the 
interest  for  $60,  and  then  double  it;  if  it  was 
$180,  we  would  multiply  by  3,  etc. 

Example  2. — Required  the  interest  of  $45  for 
219  days  at  8  per  cent. 

219r=the  days.  Here  we  cut  off  the  two 

$2  I  19  cts.  Ans.       right  hand  figures  for  cents. 

The  student  should  bear  in  mind  that  the  inter- 
est on  $45  for  219  days  is  just  the  same  as  the 
interest  on  $219  for  45  days. 

By  looking  at  the  table  we  see  that  the  interest 
on  $45  at  8  per  cent,  is  1  cent  a  day ;  hence  the 
time  in  days  is  the  answer  in  cents.     If  this  note 


112      ORTON  &  Sadler's  calculator. 

was  $'22.50,  instead  of  $45,  we  would  first  get  the 
interest  for  $45,  and  then  divide  hy  2 ;  if  it  was 
875,  we  would  add  on  § ;  if  $60,  add  on  J,  etc. 

Example  3. — Required  the  interest  of  $48  for 
115  days  at  9  per  cent. 
115— the  days.  $48=^$40+$8. 

5)$1  I  15  cts.rzithe  int.  of  $40  for  115  days. 
I  23  cts.=:the  int.  of  $8  for  115  days. 

Ans.  $1  I  38  cts.=:the  int.  of  $48  for  115  days. 

Here  we  first  find  the  interest  of  $40,  because 
ihe  days  is  the  interest  in  cents ;  then  we  divide  by 
5  to  find  the  interest  for  $8 ;  then  by  adding  both 
we  find  the  interest  for  $48,  as  required. 

Example.  4 — Required  the  interest  of  $260  foi 
104  days  at  7  per  cent. 

$52X5=$260. 
104=the  days. 

$1     04  cts=the  int.  of  $52  for  104  days. 
Am,  $5     20  cts.     Multiply  by  5. 

Here  we  first  find  the  interest  of  $52,  because 
the  days  is  the  interest  in  cents ;  then  we  multiply 
by  5  to  get  it  for  $260.  We  could  have  worked 
this  note  by  the  Bankers*  Method,  just  as  well,  by 
cutting  off  two  figures  in  the  principal,  making 
$2.60  cts.  the  interest  for  52  days,  and  then  multi- 
ply by  2  to  get  it  for  104  days.  The  student  must 
remember  that  the  interest  of  $260  for  104  days  is 
just  the  same  as  the  interest  of  $104  for  260  days 


bankers'  method  of  computing  interest.  113 

Problems  Solved  hy  Both  Methods. 

We  will  now  solve  some  examples  by  both  meth- 
ods, to  further  illustrate  this  system,  and  for  the 
purpose  of  teaching  the  pupil  how  to  use  his  judg- 
ment. He  will  then  have  learned  a  rule  more  val- 
uable than  all  others. 

Example  5.— What  is  the  interest  $180  for  76 
days  at  6  per  cent.? 

Operation  by  taking  advantage  of  the  9. 

75=the  days.  $60x3=$180. 


$0 


75  cts.=the  int.  of  $60  for  75  days. 
3  Multiply  by  3. 


Ans,   $2  I  25  cts.==the  int.  of  $180  for  75  days. 
Operation  by  the  Bankers'  Method. 
$180=:the  principal.  60da.-f  15da.=:75d», 

4)$1  I  80  cts.=:the  int.  for  60  days. 
I  45  cts.=:the  int.  for  15  days. 


Ans,  $2  I  25  cts.=the  int.  for  75  days. 

By  the  first  method  we  multiplied  by  3,  because 
3x$60nr$180;  by  the  second  method  we  added 
on  J,  because  60da.-(-  ^^da.=75da. 

N.  B. — When  advantage  can  be  taken  of  both 
time  and  principal,  if  the  student  wishes  to  prove 
his  work,  he  can  first  work  it  by  the  Bankers' 
Method,  and  then  by  taking  advantage  of  the  prin- 
cipal, or  vice  versa.  And  as  the  two  operations  arc 
entirely  different,  if  the  same  result  is  obtained  by 
«a/».h,  he  may  fairly  conclude  that  the  Kork  is  correct 


On  all  notes  that  hear  $12  per  annum^  or  any  all- 
quotpart  or  multiple  of  $12.  * 

If  a  note  bears  $12  per  annum,  it  will  certainly 
bear  $1  per  month ;  hence  the  time  in  months 
would  be  the  interest  in  $ ;  and  the  decimal  parts 
of  a  month  would  be  the  interest  in  decimal 
parts  of  a  $ ;  therefore  when  the  note  bears  $12 
per  annum  we  have  the  following  rule : 

Rule. — Reduce  the  years  to  months,  add  in  the 
^iven  monthsy  and  place  one- third  of  the  days  to  the 
right  of  this  number,  and  you  have  the  interest  in 
dimes. 

Example  1. — Required  the  interest  of  $200 
for  3  years,  7  months,  and  12  days,  at  6  per  cent. 

200  i  of  12  days=4. 

6 

Tj.  Mo.  Da. 

$12.00=int.  for  I  yr.  3     7  12=43.4mo. 

Hence  43.4  dimes,  or  $43.40ct8.,  Ans, 

We  see  by  inspection  that  this  note  bears  $12 
Uitfirest  a  year ;  hence  the  time  reduced  to  mouths, 

114 


UQHTNING  METHOD  OP  COMPUTING  INT.     115 

with  one-third  of  the  days  to  the  right,  is  the  in- 
terest in  dimes.  If  this  note  bore  $6  a  year,  in- 
stead of  $12,  we  would  take  one-half  of  the  above 
interest;  if  it  bore  $18,  instead  of  $1^,  we  would 
add  one-half  J  if  it  bore  $24,  instead  of  $12,  we 
would  multiply  by  2,  etc. 

Example  2. — Required    the   interest   of  $150 
for  2  years,  5  months,  and  13  days,  at  8  per  cent. 
150  J  of  13  days=z4J. 


8 


Yr.  Mo.  Da. 


$12.00=:int.  for  1  yr.  2    5  13r=z29.4jmo&. 

Hence  $29.4J  dimes,  or  $29.43J  cts.,  Ans. 
We  see  by  inspection  that  this  note  bears  $12 
interest  a  year ;  hence  the  time  reduced  to  months, 
with  one-third  of  the  days  placed  to  the  right,  gives 
the  interest  at  once. 

Example  3. — Required  the  interest  of  $160  for 

11  years,  11  months,  and  11  days,  at  7^  per  cent 

160  i  of  11  days=:3|. 

— Tr.  Mo.  Da 

812.00^int.forlyr.  11  11  ll=:143.3|mo8. 

Hence  $143.3f  dimes,  or  $143.36f  cts.,  Ans. 

When  the  Interest  is  more  or  less  than  $12  a  Year, 
Rule. — First  find  the  interest  for  the  given  time 
on  the  hose  o/  $12  interest  a  year ;  then^  if  the  in- 
terest on  the  note  is  only  $6  a  year,  divide  by  V^;  ij 


116      ORTON  <fe  Sadler's  calculator. 

$24  a  year^  multiply  hy  2\  if  $18  a  year^  add  on 
one-half  etc. 

Example  1. — What  is  the  interest  of  $300  for 
4  years,  7  months,  and  18  days,  at  6  per  cent. 

J  of  18  days=6. 
300  4yr.  7mo.  18da.=:55.6mo. 

6 


$1  S.OOmint.  for  1  year.     2)55.6,  int,  at  $12  a  year. 
$18=1^  times  $12.  278 


$83.4  Am. 

If  tne  interest  was  $12  a  year,  $55,^0  would  be 
the  answer ;  because  55.6  is  the  time  reduced  to 
months ;  but  it  bears  $18  a  year,  or  IJ  times  12 ; 
hence  1^  times  55.6  gives  the  interest  at  once. 

Example  2. — Required  the  interest  of  $150 
for  3  years,  9  months,  and  27  days,  at  4  per  cent. 

J  of  27  days=r9. 

150  3yr.  9nro.  27da.=:45.9mo. 

4  2)45.9,  int.  at  $12  a  year. 


$6.00r=int.  for  1  year.     $22.95  Arts. 
$6==^  times  $12. 

If  the  interest  was  $12  a  year,  $45.90  would  be 
the  answer ;  because  245.9  is  the  time  reduced  to 
months;  but  it  bears  $6  a  year,  or  J  times  12; 
lieDce  J  times  45.9  gives  the  interest  at  once. 


COMPUTING  INTEREST 


FOR  YEARS,  MONTHS,  AND  DAYS. 

The  computation  of  simple  interest,  where  the 
time  consists  of  years,  months,  and  days,  is  quite 
difficult.  Taking  the  aliquot  parts  for  the  differ- 
ent portions  of  time  almost  invariably  involves  the 
calculator  in  fractions,  and,  unless  he  is  well  versed 
in  vulgar  fractions  he  will  not  be  able  to  arrive  at 
the  correct  result.  We  have  three  bases  by  which 
we  compute  interest  at  different  rates  per  cent, 
and  by  which  we  are  enabled  to  entirely  avoid  the 
use  of  fractions.  These  three  bases  are  each  obtained 
different  from  the  oth«r,  and  consequently  we  have 
three  rules  for  computing  interest :  one  at  a  base  of 
one  per  cent.,  a  second  at  a  base  of  twelve  per 
sent.,  and  a  third  at  a  base  of  thirty-six  per  cent. 
BuLE  for  computing  interest  at  1  per  cent. : 
Take  one-third  of  the  number  of  days  and  annex 
to  the  number  of  months ;  divide  the  nimiber  thus 
formed  by  12;  annex  the  quotient  thus  obtained  to 
the  number  of  yearSy  and  multiply  the  principal  by 
this  number ;  if  the  principal  contains  centSy  point 
off  five  decimal  plac^ ;  if  not,  point  off  three  deci- 
117 


118      ORTON  &  Sadler's  calculator. 

mal  J) laces ;  this  will  give  the  interest  at  one  per 
cent.  For  any  other  rate  per  cent.,  multipli/  the  in- 
terest at  one  per  cent,  hy  the  required  rate  per  cent. 

Remark. — This  rule  applies  to  all  problems  h 
aterest  where  the  days  are  divisible  by  3,  and  thib 
number,  annexed  to  the  number  of  months,  divisi- 
ble by  12. 

EXAMPLE. 

Required  the  interest  on  $112,  at  1  per  cent.,  foi 
3  years,  3  months  and  18  days. 

SOLUTION. 

Take  one-third  of  the  number  of  days,  J  of  18 
z=:6,  annex  this  number  to  the  months  given,  36, 
divide  this  number  by  12,  36-i-12=i3,  annex  this 
number  to  the  year  gives,  33,  multiply  the  princi- 
pal by  33,  $112X33=3.69  6,  point  off  three  daci- 
mal  places,  and  we  have  the  required  interest, 
«3.69  6. 

EXAMPLE. 

Required  the  interest  on  $125  12,  at  7  per  cent., 
for  2  years,  8  months  and  12  days. 

SOLUTION. 

Take  one -third  of  the  number  of  days,  \ 
of  12=:4,  annex  this  number  to  the  number  of 
months  we  have   84,  divide  this  number   by  12, 


merchants'  method  of  computing  int.    119 

84  f-12=7,  annex  this  number  to  the  $125  12 

number  of  years  we  have  27,  multiply  27 

the    principal    by    this     number,  and       

point  off  five  decimal  places,  and  you  3.37824 

have  the  interest  at  one  per  cent.;  mul-  7 

tiply  this  interest  by  7,  and  you  have     

the  interest  at  7  per  cent.,  the  required  $23  .64768 

rate. 

EXAMPLE. 

Required  the  interest  on  $1,023,  at  8  per  cent, 
for  1  year,  9  months  and  18  days. 

SOLUTION. 

Take  one-third  the  number  of  days  and  annex 

to   the  number  of  months,  J  of   18=6,  we  have 

96-7-12=8,  annex  this  number  to  the  years       SI  023 

we  have  18,  multiply  the  principal  by  18 

this  number,  and  point  of  three  decimal     

places,  which  gives  the  interest  at  1  per  SI 8  .414 
cent.;    multiply  the  interest  at  one  per  8 

cent,  by  8,  and  you  have  the  required  in 

terest.  S147  .312 

Remark. — This  rule  will  apply  to  all  problems 
m  interest  if  one- third  of  the  number  of  the  days 
bt>  taken  decimally  and  annexed  to  the  number  of 
months,  and  this  number,  divided  by  12,  carried 
out  decimally.  But  this  makes  the  multiplier 
very  large  ;  hence,  to  avoid  this  large  number  id 


120      ORTON  &  Sadler's  calculator. 

the  multiplier,  where  the  days  are  divisible  by  3, 
and  this  number,  annexed  to  the  months,  is  not 
divisible  by  12,  we  use  the  following  rule,  called 
our  base  at  12  per  cent. : 

KuLE. — Reduce  the  years  to  months^  add  in  the 
months^  take  one- third  of  the  number  of  days  and 
annex  to  this  number^  multiply  the  principal  by  the 
number  thus  formed ;  if  there  are  cents  in  the  prin- 
cipal^ point  off  five  decimal  places  ;  if  there  are  no 
cents  in  the  principal^  point  off  three  decimal  places  ; 
this  gives  the  interest  at  12  per  cent.  For  any  other 
rate  per  cent.^  take  such  part  of  the  base  before  mul- 
tiplying as  the  required  rate  is  a  part  of  12, 

EXAMPLE. 
Required  the  interest  on  $123,  at  12  per  cent 
for  2  years,  2  months  and  six  days. 

SOLUTION. 

Reduce   the   2   years   to    months   gives   us   24 
months,  add  on  the  2  months  gives  us  26 
aionths,  take  one-third  of  the  days,  J  of         $123 
6=2.,  annexed   to   the  26  months   gives  262 

262,  which  constitutes  the  base ;  multiply     — 

the  principal  by  this  base,  and  you  have  $32  .226 
the  interest  at  12  per  cent. 

EXAMPLE. 

Required  the  interest  on  $144,  at  6  per  cent.,  for 
4  years,  5  months  and  12  days. 


MERCHANTS*  METHOD  OP  COMPUTING  INT.     121 

SOLUTION. 

Reduce  the  4  years  to  months  gives  48  months, 
add  in  the  5  months  gives  53  months,  take  one- 
third  of  the  days  and  annex  to  the  number  of 
months,  J  of  12=4.  annex  to  the  53  months,  534 ; 
this  number  multiplied  into  the  principal  would 
give  the  interest  at  12  per  cent.  But  we  want  it 
at  6  per  cent.  We  will  now  take  such  part  of 
either  principal  or  base  as  6  is  a  part  of  12 ;  6  is 
J  of  12,  therefore  we  will  take  J  of  144=72 
otie-half  of  the  principal,  and  mul-  534 

tiply  it   by  the  base,  which  will  

give  the  interest  at  6  per  cent,  $38,448 

EXAMPLE. 

Required  the  interest  on  $347  25,  at  8  per  cent., 
for  2  years,  3  months  and  9  days. 
SOLUTION. 

Reduce  the  2  years  to  months,  24  months,  add 
the  3  months,  27  months,  take  one-third  of  the 
days,  J  of  9=3,  annex  to  th^  months,  273,  the 
base;  this,  multiplied  into  the  principal,  would 
give  the  interest  at  12  per  cent.  But  we  want  the 
interest  at  8  per  cent ;  we  will  take 
two-thirds  of  the  base  before  multiply-  $347  25 
ing!    f   of    273=182;    the    principal  182 

multiplied  by  this  number  gives  the 

interest  at  8  per  cent.  $63.19950 

Remark. — This  base  is  used  where  the  days  are 

divisible  by  3,  and  the  number  formed  by  annex- 
11 


122   ORTON  &  Sadler's  calculator. 

ing  one-third  of  the  days  to  the  months  not  divisi  • 
ble  by  12.  We  now  come  to  time  in  which  neithei 
days  nor  months  are  divisible.  Where  such  time 
as  this  occurs,  we  use  a  base  at  36  per  cent. 

Rule. — Reduce  the  time  to  dai/s,  hy  multiplying 
the  years  hy  12,  adding  in  the  months^  if  any^  and 
multiplying  this  number  6y  30,  adding  in  the  days, 
if  any;  multiply  the  principal  hy  this  number^ 
pointing  off  5  decimal  places^  where  cents  are  given 
in  the  principal,  and  3  places  where  no  cents  are 
given.     This  will  give  the  interest  at  36  per  cent 

EXAMPLE. 
Required  the  interest  on  $144,  at  36  per  cent., 
for  3  years,  2  months  and  2  days. 
SOLUTION. 
Reduce  the  time  to  days  gives   1142         $144 
days ;  multiply  the  principal  by  this  base,         1142 

and   you  have    the    interest    at   36    per  - 

cent  $164,448 

EXAMPLE. 

Required  the  interest  on  $144,  at  9  per  cent.,  lor 
5  years,  7  months  and  5  days. 
SOLUTION. 

Reduce  the  time  to  days  gives  2,015  days;  if 
we  multiply  the  principal  by  this  base,  we  would 
get  the  interest  at  36  per  cent.;  but  we  want  it  at 
.9   pei    cent.     We  can  ^ake  such  part  of   either 


MERCHANTS'  METHOD  OF  COMPUTING  INT.    123 

principal  oi  base  as  9  is  a  part  of  36  before  multi- 
plying ;  9  is  J  of  36 ;  wo  will  take  J  of  the  prin- 
cipal, it  being  divisible  by  4  ;  J  of  144=36,  2915 
which,  multiplied  into  the  base,  will  give  36 
the  interest  at  9  per  cent.,  by  pointing  ofi  

3  decimal  places.  S72.540 

EXAMPLE. 

Bequired  the  interest  on  $875  15,  at  6  per  cent.» 
for  5  years,  7  months  and  12  days. 
SOLUTION. 

Reduce  the  time  to  days  gives  2022  days ;  6  is 
J  of  36 ;  take  one  sixth  of  the  base, 
i  of  2022=337;  multiply  the  prin-         $875  15 
cipal  by  this  number,  point  off  5  dec-  337 

imal  places,  and  you  have  the  interest 

at  6  per  cent.,  the  required  rate.  $294.92555 

Remark. — We  have  now  fully  explained  our 
method  of  computing  interest  at  the  three  different 
bases.  Any  and  every  problem  in  interest  can  be 
solved  by  one  of  these  three  bases.  Some  prob- 
lems can  be  solved  easier  by  one  base  than  another. 
Where  the  days  are  divisible  by  3,  and  their  num- 
ber, annexed  to  the  months,  divisible  by  .12,  it  is 
the  shortest  and  best  method  to  use  the  base  at  1 
per  cent.  By  using  one  or  the  other  of  these  three 
bases,  the  student  can  avoid  the  use  of  vulgar 
fractions.  The  student  must  study  these  three 
principles  carefully,  and  learn  to  adopt  readily  the 
base  best  suited  to  the  problem  to  be  solved. 


To  compute  interest  on  notes,  bonds,  and  mon 
gages,  on  which  partial  payments  have  been  made, 
two  or  three  rules  are  given.  The  following  is 
called  the  common  rule,  and  applies  to  cases  where 
the  time  is  short,  and  payments  made  within  a  yeai 
cf  each  other.  This  rule  is  sanctioned  by  custom 
and  common  law;  it  is  true  to  the  principles  of 
simple  interest,  and  requires  no  special  enactment. 
The  other  rules  are  rules  of  law,  made  to  suit  such 
cases  as  require  (either  expressed  or  implied)  an- 
nual interest  to  be  paid,  and  of  course  apply  to  no 
business  transactions  closed  within  a  year. 

Rule. —  Compute  the  interest  of  the  principal  sum 
for  the  whole  time  to  the  day  of  settlement,  and  find 
the  amount.  Compute  the  interest  on  the  several  pay- 
ments, from  the  time  each  was  paid  to  the  day  oj 
settlement ;  add  the  several  payments  and  the  vater- 
est  on  each  together,  and  call  the  sum  the  amount  oJ 
the  payments.  Subtract  the  amount  of  the  payments 
from  the  a/mount  of  the  principal,  will  lextvc  the  sitm 
due, 

124 


PAETIAL    PAYMENTS.  125 

EXAMPLES. 
I.  A  gave  his  Dote  to  B  for  $10,000 ;  at  the  end 
0^4  months,  A  paid  $6,000;  and  at  the  expiration 
of  another  4  months,  he  paid  an  additional  sum  of 
$3,000 ;  how  much  did  he  owe  B  at  the  close  of  the 
year? 

By  the  Oommon  Rule, 

Principal $10,000 

Interest  for  the  whole  time 600 


Amount $10,600 

1st  payment $6,000 

Interest,  8  months      240 

2d  payment 3,000 

Interest,  4  months        60 


Amount $9,300  9,300 


Due $1300 

PROBLEMS   IN   INTEREST. 

There  are  four   parts  or  quantities  connected 
with  each  operation  in  interest :  these  are,  the 
Principal,  Mate  per  cent,  Time,  Interest  or  Amount 

If  any  three  of  them  are  given  the  other  may  be 
found. 

Principal,  interest,  and  time  given,  to  find  the 
rate  per  cent. 

1.  At  what  rate  per  cent,  must  $500  be  put  on 
interest  to  gain  $120  in  4  years  ? 


126      ORTON  &  Sadler's  calculator. 

Operation.  Bj  analysis 

8500  The   interest  of 

.01  $1    for    the    given 

— -  time  at  1  per  cent. 

6.00  is   4   cents.     8500 

4  will    be  500  times 

asmuch=:500X.04 

20.00)120.00(6  per  cent.,  Ans.  =$20.00.    Then  if 

120.00  $20  give  1  per  cent., 


$120  will  give  ^^jj^ 

=z6  per  cent. 

Rule. — Divide  the  given  interest  hy  the  interest 

of  the  given  sum  at  I  per  cent,  for  the  given  time, 

and  the  quotient  will  be  the  rate  per  cent,  required 

Principal,  interest,  and  rate  per  cent,  given,  to 

find  the  time. 

2.  How  long  must  $500  be  on  interest  at  6  per 
cent,  to  gain  $120  ? 

Operation  By  analysis. 

$500  We  find  the  in- 

.06  terest  of  $1.00   at 

the  given  rate   for 

30.00)120.00(4  years,  Ans.  1  year  is  6  cents 
120.00  $500,  will  therefore 
be    500    times    as 


much=500X  .06=:$30.00.  Now,  if  it  take  1  year 
to  gain  $30,  it  will  require  ^V*  to  gain  $120=4 
years,  Ans, 


PARTIAL    PAYMENTS.  127 

Rule. — Divide  the  given  interest  hy  th^  interest 
of  the  principal  for  1  ^ear,  and  the  quotient  is  the 
time. 

Given  the  amount,  time,  and  rate  per  cent.,  to 
find  the  'principal. 

Rule. — Divide  the  given  amount  hy  the  amount 
o/"?!,  at  the  given  rate  per  cent.,  for  the  given  time. 

Remark. — This  rule  is  deduced  from  the  fact 
that  the  amount  of  different  principals  for  the  same 
time  and  at  the  same  rate  per  cent.,  are  to  each 
other  as  those  principals. 

BANK    DISCOUNT. 

Bank  Discount  is  the  sum  paid  to  a  bank  for  the 
payment  of  a  note  before  it  becomes  due. 

The  amount  named  in  a  note  is  called  the  face 
of  the  note.  The  discount  is  the  interest  on  the 
face  of  the  note  for  3  days  more  than  the  time 
specified,  and  is  paid  in  advance.  These  3  days 
are  called  days  of  grace,  as  the  borrower  is  not 
obliged  to  make  payment  until  their  expiration. 
Hence,  to  compute  bank  discount,  we  have  the  fo^ 
lowing 

Rule. — Find  the  interest  on  the  face  of  the  note 
for  3  days  more  than  the  time  specif ^d ;  this  will 
he  the  discount.  From  the  face  of  the  note  deduct 
the  discount,  and  thi  remainder  will  be  the  PRESENT 
VALUE  of  (he  note. 


128      ORTON  &  Sadler's  calculator. 

DISCO riNT,    OR   COUNTING    BACK. 

The  object  of  discount  is  to  show  us  what  al« 
lowance  should  be  made  when  any  sum  of  money 
Is  paid  before  it  becomes  due. 

The  present  worth  of  any  sum  is  the  principal 
that  must  be  put  at  interest  to  amount  to  that  sum 
in  the  given  time.  That  is,  $100  is  the  'present  worth 
of  $106  due  one  year  hence;  because  SlOO  at  6  per 
cent,  will  amount  to  $106  j  and  $6  is  the  discount, 

1.  What  is  the  present  worth  of  $12.72  due  one 
year  hence  ? 

First  method.  Second  method. 

$12.72  $ 

100  1.06)12.72($12  Am. 

10.6 

106)1272.00($12  Am,  

106  2.12 

2.12 

212  

212 

As  $100  will  amount  to  $106  in  one  year  at  6 
per  cent.,  it  is  evident  that  if  |gg  of  any  sum  be 
taken,  it  will  be  its  present  worth  for  one  year,  and 
that  ygg  will  be  the  discount.  And  as  $1  is  the 
present  worth  of  $1.06  due  one  year  hence,  it  r 
evident  that  the  present  worth  of  $12.72  must  be 
equal  to  the  number  of  times  $12.72  will  contain 

n.06. 


EQUATION   OP   PAYMENTS.  129 

Rule. — Divide  the  given  sum  hy  the  amount  of 
A  for  the  given  rate  and  timCy  and  the  quotient  wilt 
>e  the  present  worth.     If  the  present  worth  be  sub- 
tracted from  the  given  sum,  the  remainder  will  be  the 
discount. 


ijEDUATlDNOFPAYMENTSi 


Equation  op  Payments  is  the  process  of  find- 
ing the  equalized  or  average  time  for  the  payment 
of  several  sums  due  at  different  times,  without  losa 
to  either  party. 

To  find  the  average  or  mean  time  of  payment, 
when  the  several  sums  have  the  same  date. 

Rule. — Multiply  each  payment  by  the  time  thai 
must  elapse  before  it  becomes  due;  then  divide  the 
sum  of  these  products  by  the  sum  of  the  payments, 
and  the  quotient  will  be  the  averaged  time  required. 

Note. — When  a  payment  is  to  be  made  down,  it 
has  no  product,  but  it  must  be  added  with  the 
other  payments  in  finding  the  average  time. 

Example  1. — I  purchased  goods  to  the  amount 
of  $1200;  $300  of  which  I  am  to  pay  in  4  months, 
$i00  in  5  months,  and  $500  in  8  montls.  IIow 
long  a  credit  ought  1  to  receive,  if  1  pay  th* 
wbolo  sum  at  once?  -4ns.  6  uoaths. 


130   ORTON  &  Sadler's  calculator. 

Mo.  Mo.  r     A  credit  on  $300  for*  xflDnthsIl 

^vyOAA       1  OAA  •<  the  same  as  the  credit  on  $1  foi 

4X'3UU=1ZUU  h200  months.       ' 

i"    A  credit  on  8400  for  6  months  ii 
5<400r=:2000  -{the  same  as  the  credit  oa  81  for 

<  2000  months. 

Qv^  r  AA Af\(\f\  \     A  credit  on  8500  for  8  months  is 

O/^UUU ^Kjyjyj  ^  the  same  as  the  credit  on  $1  for 

—      (4000  months. 

1 0AA\  TOAA  /a  Therefore,   I    should    have    th© 

IZUUj  <ZUU  {p  mo.       same  credit  as  a  credit  on  $1  for 

rji^r\(\  7200    months,   and  on  $1200,  the 

**^""  whole  sum,  one-twelfth  hundredth 

___  part  of  7200  months,   which  is  6 

months. 

This  rule  is  the  one  usually  adopted  by  mer- 
chants, although  not  strictly  correct,  still,  it  is  suf- 
ficiently accurate  for  all  practical  purposes^ 

To  find  the  average  or  mean  time  of  payment, 
when  the  several  sums  have  different  dates. 

Example  1. — Purchased  of  James  Brown,  at 
sundry  times,  and  on  various  terms  of  credit,  as  by 
the  statement  annexed.  When  is  the  medium  time 
of  payment? 

Jan.      1,  a  bill  am'ting  to  $360,  on  3  months*  credit. 
Jan.    15,  do.       do.  186,  on  4  months' credit. 

March  1,  do.       do.  450,  on  4  months' credit. 

May    15,  do.       do.  300,  on  3  months' credit 

June  20,  do.       do.  500,  on  5  months'  credit. 

Ans,  July  24th,  or  in  115  da. 
Due  April    1,  $360 

May   15,    186X  44=    8184 

July     1,    450X   91=  40950 

Aug.  15,    300X136=  40800 

Nov.  20,    500X233=116500 

1796V  into  ) 206434 (114gj|  days. 


EQUATION    OF   PAYMENTS.  131 

We  first  find  the  time  wheo  each  of  the  bills 
mil  become  due.  Then,  since  it  will  shorten  the 
operation  and  bring  the  same  result,  we  take  the 
time  when  the  first  hill  becomes  due,  instead  of  its 
date,  for  the  period  from  which  to  compute  the 
average  time.  Now,  since  April  1  is  the  period 
from  which  the  average  time  is  computed,  no  time 
will  be  reckoned  on  the  first  bill,  but  the  time  for 
the  payment  of  the  second  bill  extends  44  days  be- 
yond April  1,  and  we  multiply  it  by  44. 

Proceeding  in  the  same  manner  with  the  remain- 
ing bills,  we  find  the  average  time  of  payment  to 

114  days  and  a  fraction,  from  April  1,  or  on  the 
24th  of  July. 

Rule. — Find  the  time  when  each  of  the  sums  be- 
comes due,  and  multiply  each  sum  by  the  number  of 
days  from  the  time  of  the  earliest  payment  to  the 
payment  of  each  sum  respectively.  Then  proceed  as 
in  the  last  rule,  and  the  quotient  will  be  the  aver- 
age time  required,  in  days,  from  the  earliest  pay- 
ment. 

Note. — Nearly  the  same  result  may  be  obtained 
by  reckoning  the  time  in  months. 

In  mercantile  transactions  it  is  customary  to  give 
a  credit  of  from  3  to  9  months,  on  bills  of  sale. 
Merchants  in  settling  such  accounts,  as  consist  of 
various  items  of  debit  and  credit  for  different  times^ 
generally  employ  the  following: 


132   ORTON  &  Sadler's  calculator. 

Rule. — Place  on  the  debtor  o'**  credit  side,  suck  a 
mm,  (which  may  he  called  merchandise  balance,) 
as  will  balance  the  account 

Multiply  the  number  of  dollars  in  each  entry  by 
the  number  of  days  from  the  time  the  entry  was  madt 
to  the  time  of  settlement;  and  the  Merchandise  bal- 
ance by  the  number  of  days  for  which  credit  was 
given.  Then  multiply  the  difference  between  the  sum 
of  the  debit,  and  the  sum  of  the  credit  products,  by 
the  interest  of  $1  for  1  day ;  this  product  will  be 
the  interest  balance. 

When  the  sum  of  the  debit  products  exceed  the  sum 
of  the  credit  products,  the  interest  balance  is  in  favor 
of  the  debit  side ;  but  when  the  sum,  of  the  credit 
products  exceed  the  sum  of  the  debit  products,  it  is  in 
favor  of  the  credit  side.  Now  to  the  merchandise 
balance  add  the  interest  balance,  or  subtract  it,  as  the 
case  may  require,  and  you  obtain  the  cash  BAL- 
ANCE. 

A  has  witli  B  the  following  account : 


1849. 

Dr. 

1849.                                          Or. 

Jan.  2.     To  merchandise, 

$200 

Feb.  20.  By  merchandise,   $100 

April  20.  •♦           ♦* 

400 

May.  10.  ♦*               ♦•                300 

If  interest  is  estimated  at  7  per  cent.,  and  a 
yredit  of  60  days  is  allowed  on  the  different  sums, 
what  is  the  cash  balance  August  20,  1849? 

Am.  206.54. 

Explanation. — Without  interest  the  cash  bal 
ance  would  be  $200. 


EQUATION    OF   PAYMENTS.  133 

If  no  credit  had  been  given,  the  debits  should 
be  increased  by  the  interest  of  $200  for  230  days, 
at  7  percent.;  and  the  interest  of  $400  for  122 
days,  at  7  per  cent.  The  credits  should  be  increas- 
ed by  the  intere^  of  $100  for  181  days,  at  7  per 
cent.,  and  the  interest  of  $300  for  102  days,  at  7 
per  cent. 

Since  a  credit  of  60  days  is  given  on  all  sums,  it 
is  evident  by  the  above  calculation,  that  we  should 
increase  the  debits  by  the  interest  of  the  sum  of 
the  debits,  $600,  for  60  days  more  than  justice  re- 
quires. Also,  that  we  should  increase  the  credits 
by  the  interest  of  the  sum  of  the  credits,  $400,  for 
60  days  more  than  we  should  do. 

Now,  instead  of  deducting  these  items  of  inter- 
est from  the  amount  of  debit  and  credit  interests, 
it  is  plain  that  it  will  be  more  convenient  and 
equally  just,  to  diminish  the  debit  interest  of  the 
merchandise  balance  for  60  days,  which  can  be 
most  readily  accomplished  by  adding  the  interest 
on  the  merchandise  balance  for  60  days,  to  the 
credit  items  of  interest. 

From  which  we  discover  that  the  interest  balance 
is  equal  to  the  difference  between  the  sum  of  the 
debit  interests,  and  the  sum  of  the  credit  intereste 
increased  by  the  interest  of  the  merchandise  bal- 
ance for  the  time  for  which  credit  was  given. 
12 


134      ORTON  &  Sadler's  calculator. 
Operation. 

DEBITS.  CREDITS. 

f        Days.  $        Days. 

200X230=46000  100x181=18100 

400X122=48800  300X102=30600 

Balance,  200X   60=12000 


94800 
60700 


60700 


0.07 

) 

365 


-X 34100=86.54  Interest  halance^  yearly. 


Therefore,  the  foregoing  account  becomes  bal- 
anced as  follows : 


1849.  Dr. 

fan.     2.  To  Merchandise,  $200.00 
\pril20.  "  •*  400.00 

/kug.  20.  "  balance  of  int.       6.54 


$006.54 


1849  Or, 

Feb.  20.  By  Merchandise,  $100.00 
May.  10.  ♦'  "  300.00 

Aug.  20.  "  balance,  206.54 


$606.54 


A-Ug.  20.  *•  Cash  balance,  $206.64 

Note. — It  is  customary  in  practice,  when  the 
number  of  cents  in  any  of  the  entries,  are  less  than 
50,  to  omit  them,  and  to  add  $1  when  they  are  66, 
or  more. 


LIGHTNING    METHOD 
or 

AVERAGING    ACCOUNTS. 

Trom  •  PACKARD'S  KEY  TO  COMPLETE  COURSE."    With  kiud 

parmissioi)   of  the  author,   S.  S.   Packard,  Presideut  of 
I*iickard's  New  York  Business  College. 

The  matter  of  averaging  accounts,  or  which 
is  usually  styled  in  the  arithmetics,  "  Compound 
Average,"  is  of  such  immediate  and  vital  impor- 
tance to  the  accountant,  that  we  submit  here  a 
short  method  which,  from  its  mechanical  advan- 
tages and  other  considerations,  has  beeu  well-. 
received  by  those  who  have  had  occasion  to  use 
it.  For  reasons  that  will  be  apparent,  it  is  not 
theoretically  exact,  but  the  discrepancies  are  so 
135 


136      ORTON  &  Sadler's  calculator. 

small  and  unimportant  that  they  are  scarcely 
worthy  a  thought,  practically  speaking,  and 
besides  as  a  fraction  of  a  day  in  the  results  of 
average  cannot  be  considered,  the  slight  inciden- 
tal variations  in  this  method  have  no  real  bearins: 
on  the  result. 

This  method  has  been  in  use  for  the  past 
fifteen  or  twenty  years,  and  its  main  features 
have  been  j^ublished  by  different  authors;  and 
yet  certain  itinerant  professors  are  in  the  habit 
of  claiming  its  originality,  and  offering,  under 
pledge  of  secrecy,  and  for  a  substantial  con- 
sideration in  hand  paid,  to  disclose  its  wonderful 
properties. 

It  is  neither  more  nor  less  than  a  convenient 
mechanical  arrangement,  whereby  the  principles 
of  average  are  effectively  and  compactly  enforced. 

We  will  illustrate  its  workings  from  the  ma- 
terials contained  in  the  following  example : 

EXAMPLE. 

When  is  the  balance  of  the  following  account 
due? 

S.   S.   PACKARD. 


1871                Debits. 
May  12 750 

♦'     30 117 

June  12 340 

July  1 150 


1871  Credits. 

June  10 500 

"      30 300 

Due  by  average;  April  26. 


AVERAGING   ACCOUNIS.  137 

ILLUSTRATION. 

Note. — The  assumed  date  is  fixed  on  the  31st  of  Decem- 
ber, preceding  the  earliest  date  in  the  account.  This  is 
for  convenience  sake,  and  to  preserve  constant  uniformity. 
Some  prefer  the  last  day  of  the  month  preceding  the  first 
item.  Interest  is  reckoned  on  each  amount  from  Decem- 
ber 31  to  the  date  due  at  \1%  per  annum,  or  1%  per 
month.  This  rate  of  percentage  or  interest  is  also  arbi- 
trarily fixed,  as  being  the  most  convenient  for  use. 

Debits.  PROCESS.  Credits. 


Maj  12...Sio0j   „    i2day8,  3  00 
"     30.... 117  (   "    5rao3.,   5  85 


June  12.... 340-^ 


5  mos.,  17  00 
12  days,    1 


Inlvl  1')0i    "    6n>os.,   9  00 

•'"'^  ^ ^''"i    "    Iday,         05 


JunelO...S500|  »  iq  days,    1  67 

"      30....  300  j  "    6  mos.,  18  00 

Total  Cr.  J^  "800    Tot  Cr.Int.  41  CT 


Total  Dr.")  /Total  Dr.") 

of  *      j  1357)      of  Int.  i66  26 
Total  Cr.  )  /  Total  Cr.  ) 

of  ^   /  800  \     of  Int.  J  44  67 

Bal.  of  *  ~lui    Bal.  of  Int.  21  59(3  months. 
16  71 
4  88 
30 

14640(26  + days. 
1114 


3500 

3343 

158 


Ans. — Balance  of  5^  due  in  3  mos.,  26  days,  from  Dec.  31,  or  April  26. 

We  regret  that  our  limited  space  will  not  permit  a  more 
detailed  exhibit  of  the  method,  either  as  to  its  philosophy 
or  its  facts.  The  above  working  must  carry  its  own  sug- 
gestions. The  theory  is,  that  if  a  settlement  were  made 
as  on  the  preceding  31st  of  December  (the  assumed  day 
of  settlement),  the  debit  side  of  the  account,  as  shown, 
would  be  entitled  to  $66.26  discount,  and  the  credit  side 
to  $44.67,  making  a  balauce  of  $21.59  in  favor  of  the  debit 


138   ORTON  «&  Sadler's  calculator. 

side.  As  the  balance  of  account  is  also  in  favor  of  the 
debit  side,  it  is  only  necessary  to  know  how  long  it  would 
take  the  balance  of  account  to  produce  the  balance  of  in- 
terest (at  the  rate  of  1%  a  month,  or  12%  per  annum*) ;  to 
know  the  time — reckoned  forward — when  the  balance  of  ac- 
count falls  due.  Now,  as  the  rate  named  (1%  a  month), 
the  interest  for  one  month  can  be  had  by  merely  cutting 
off  two  figures  from  the  right  of  dollars,  we  have  the  bal- 
ance of  account  thus  divided  ($5.57),  a  ready  divisor  of 
the  balance  of  interest  ($21.59),  the  quotient  being  the 
number  of  months  and  parts  of  a  month  it  will  take  the 
balance  of  account  to  produce  the  balance  of  interest. 
This  time  reckoned  forward  from  the  assumed  focal  date, 
will  get  the  average  date  of  payment. 

Thus,  3  months,  26  days  from  December  31  will  bring 
the  average  date  as  stated — April  26th. 


*FORMULA. 

For  calculating  Interest  at\2^o  per  annum. 

Time — Months  and  days. 

Multiplied  by  the  1  __Interest  expressed 
The  Principal  J  number  of  months.  J  in  cents. 


Dollars  only.  ]     Multiplied  by  i   )  __Interest  expressed 

the  number  of  days.  J  in  mills. 

J  the  Principal  f  Multiplied  by  any  |  __Interest  expressed 
Dollars  only.    (    number  of  days.  J  in  mills. 

Note. — In  the  above  application,  when  the  j)rincipal 
contains  cents,  point  off  two  additional  decimal  places. 


^^=- 


<V^ 


Is  the  association  of  two  or  more  persons  under 
a  copartnership  name,  for  the  purpose  of  trans- 
acting business  for  their  mutual  profit,  with  cer- 
tain agreements  regarding  the  investment  of 
capital  and  the  division  of  gains  or  losses 
between  them. 

The  Partners  are  the  jlersons  associated  to- 
gether in  business. 

The  Capital  or  Stock  is  the  cash  or  property 
invested. 

The  Resources  of  a  firm  or  copartnership  is 
the  property  owned  by  them,  including  money 
and  claims  due  from  others. 

The  Liabilities  are  the  debts  owed  by  the  firm 
or  claims  against  the  copartnership. 

Net  Capital  is  the  excess  of  resources  over 
the  liabilities. 

Net  Insolvency  is  the  excess  of  liabilities  over 
the  resources. 

Net  Gain  is  the  excess  of  gains  over  the  losses 
and  expenses,  and  is  shown  by  taking  the  differ- 
ence between  the  net  capital  at  commencing 
business  from  that  shown  at  closing  the  books, 
which  is  termed  Present  Worth, 
139 


140   ORTON  &  Sadler's  calculator. 

Net  Loss  is  the  excess  of  losses  over  the  gains 
or  profits. 

DIVISION    OF  GAINS  OR    LOSSES 

Adjusted  between  parties  according  to  capital  invested. 

In  the  adjustment  of  gains  or  losses  between 
copartners  on  the  basis  of  capital  invested,  we 
have  three  methods,  i,  e.y  Percentage,  Fractions^ 
and  Proportion,  each  producing  in  the  aggregate 
the  same  result. 

FIRST   METHOD — BY    PERCENTAGE. 

Rule  I. — Ascertain  the  per  cent,  of  gain  or 
loss  (on  capital  invested)  by  dividing  the  net  gain 
or  loss  by  the  net  capital.  For  each  partner^s 
share  of  gain  or  loss,  midtiply  his  net  investment 
by  the  percentage  thus  obtained,^ 

SECOND   METHOD BY    FRACTIONS. 

Note. — As  there  will  be  as  many  fractional  parts  in  the 
adjustment  as  there  are  partners,  we  produce  the  following : 

Rule  II. —  Take  the  net  investment  of  each 
partner  for  the  numerator  of  a  fraction,  and  the 
net  capital  for  its  denominator  ;  and  for  each  part- 
ner's share  take  his  respective  fractional  part  of  the 
entire  gain  or  Zoss.* 

*  Each  partner's  share  of  gain  or  loss  added  together 
equals  the  total  gain  or  loss. 


PARTNERSHIP.  141 


THIRD    METHOD — OR   PROPORTION. 

Note. — This  method  we  consider  the  most  simple  and 
practical,  and  advise  its  use  in  preference  to  either  of  the 
above  methods,  except  in  simple  cases  of  adjustment, 
when  the  division  can  be  shown  by  small  fractions,  thus, 
i+i+i=the  total  interest  of  partners. 

Rule  III. — State  by  proportion,  as  the  total 
capital  is  to  each  partner's  investment,  so  is  the  net 
gain  or  loss  to  each  partner^ s  respective  share  of 
same,'^ 

Proper  form  of  proportional  statement, 

Tota!  capital  "I    .    f  Kach   part- 1   .  .  Net  gain  "I  .  Each  partner's 
oi  partners.  J    *  \  ner's  share,  j  *  *    or  loss.   J   '     gain  or  loss. 

Working. — Multiply  each  partner's  share  by 
the  net  gain  or  loss,  divide  that  product  by  the 
total   capital,   and    the   quotient  will    be   each 
partner's  respective  share  of  gain  or  loss. 
EXAMPLE. 

E.  Burnett,  W.  H.  Devon,  and  J.  K.  Hopper 
are  copartners,  the  gains  or  losses  arising  from 
business  to  be  divided  between  them  in  propor- 
tion to  average  investment.  The  investments 
are  as  follows:  Mr.  B.  $5500;  Mr.  D.  $6750; 
Mr.  H.  $3250.  Upon  closing  the  books  the  net 
gaiD  is  found  to  be  $3850 — what  is  each  partner's 
respective  share?  Ans.  Mr.  B.  $1366.13;  Mr. 
D.  $1676.61 ;  Mr.  H.  $807.26. 

*  Each  partner's  share  of  gain  or  loss  added  together 
equals  the  total  gain  or  loss. 


142      ORTON  &  Sadler's  calculator. 

Solution, 


B.  investment,  5500 

D. 

H. 


?2?:;iTotal  am't.  .    f  g'  ^^Tf^--  '^fJ  . .  /Net  gain) 

5500x385O=:21175000-j-1550a=1366.13     B's  share. 
6750x3850--25987500-T-1550a=1676.61     D's       '• 
3250x3850=  }251200-t-155O0=  807.26     ll's       " 
Total  gain,  $3850.00— or  Proof. 

DIVISION  OF  GAINS  OR  LOSSES 
Between  partners,  according  to  investment,  when 
the  capital  is  furnished  at  different  dates. 

Rule. — Multiply  each  partner's  investment  by 
the  time  employed.  The  product  thus  obtained 
equals  the  average  investment  for  the  average  time^ 
and  the  sum  of  the  products  the  total  average  capi- 
tal for  the  average  time. 

Note. — After  ascertaining  the  average  capital  and  in- 
vestment by  the  above  rule,  to  secure  each  partner's  re- 
spective share  of  the  gain  or  loss,  proceed  according  to 
either  rule  on  pages  140,  141. 

EXAMPLE. 
A  and  B  are  partners,  gains  or  losses  to  be 
divided  according  to  average  investment. 


A  putsin  Jan.  1,  $5000 
"  Feb.l,  1000 
"     Sep.  1,    2000 


B  puts  in  Jan.  1,  S2000 
"  Apr.l,  3000 
"     Julyl,    1000 


January  1,  one  year  from  date  of  first  invest- 
ment, the  books  are  closed,  and  the  net  gain 
ascertained  to  be  S2720 — what  is  each  partuer^s 
share?     Ans,  A's  $1580 ;  B's  $1140. 


PARTNERSHIP,  143 

Process  of  Solution, 

Invest-  Time,  Average  Average 
merit,     mos.     Capital.     Time. 

A  invested  Jan,  1  to  Jan.  1,  5000X12=^60000—1  mo. 
Feb.l'^         "      1000X11=11000—1 
Sep.  1**         "     2000  X  4=-  8000— 1 

A's  aver,  investm't  for  the  aver,  time,  79000 — 1 

B  invested  Jan,  1  to  Jan.  1,  2000X12=24000—1 
Apr.  1"  ''  3000  X  9=27000—1 
Julyl''         ''     1000 X  6=  6000—1 

B's  aver,  investm't  for  the  aver,  time,  57000 — 1 
A's     "  "  -  *'         ^*       79000—1 

A  &  B's  average  investment=Total 

capital  for  average  time,  136000 — I 

Stcdenieiit  per  Ruley  page  141. 

Total  aver.  Each  partner's      Total  Each  partner's 
capital.       aver,  capital.        gain,  share  of  gain. 

136000  :  79000  :  :  2720=1580— A's 
136000  :  57000  :  :  2720=1140— B's 


Worhing, 

[)0-^1360( 
00—1 360( 

Total  gain,  $2720  Proof. 


79000  X2720=214880000-j-l  36000=$!  580  A's  share 
67000X2720=155040000—136000=  1140  B's     " 


WHEN    GOLD    COMMANDS    A    PREMIUM. 

Rule. — Midtiply  the  amount  hy  100,  increased  by 
the  rate  per  cent,  of  premium,  and  point  off  four 
decimal  places,  if  there  are  cents  in  the  amount ;  if 
no  cents  occur,  point  off  but  two  places. 

Example.  —  What  amount  in  currency  will 
$666. 66|  gold  purchase,  when  at  a  premium  of 
50  per  cent.  ? 

Process.  — 100  +     50  =      150. 

$666,661  X  150  =  $1000  currency. 

CURRENCY  TO  GOLD. 

WHEN    GOLD    COMMANDS    A    PREMIUM. 

Rule. — Divide  the  amount  by  100  increased  by  the 
rate  per  cent,  of  premium,  and  point  off  two  less  than 
the  number  of  places  in  the  dividerid. 

Example. — What  amount  in  gold  will  $1000  cur- 
rency purchase,  gold  commanding  a  premium  of 
50  per  cent.  ? 

Process. —100  +  50  ==  150. 

$1000  -T-  150  =.:  $666.66f  gold 

150  )  1000.0,0,0,0  (  666.66f 
900 

1000 
900 

1000 
900 

1000 
900 


100 


144 


MATURITY   OF   COMMERCIAL   PAPER.      145 

TO    ASCERTAIN    THE    GOLD    VALUE    OF    CURRENCY 
WHEN    AT    A    DISCOUNT. 

Rule. — MuUi'ply  the  amount  hy  the  rate  'per  cent, 
of  discount ;  from  xuliich  subtract  the  result,  and  you 
will  have  the  sum  required. 

Example. — When  currency  is  at  a  discount  of  33^ 
per  cent.,  what  sum  in  gold  will  $1000  purchase  ? 
Process.-—  $1000   X  33^  =  discount. 
1000  —  discount  =  sum. 
1000  1000 

33i  333.33 

$333.33  $666.67 

$1000  in  currency,  at  33J  per  cent,  discount,  will 
purchase  $666.67  gold. 

Note. — When  gold  commands  a  premium  of  50 
per  cent.,  U.  S.  currency  is  at  33^  per  cent,  dis- 
count. 


MATUEITY  OF  COMMEEOIAL  PAPER. 

To  ascertain  the  Maturity  of  Commercial  Paper 
payable  in  days  after  date. — Set  down  the  days  in 
full ;  from  which  take  the  number  of  days  remain- 
ing in  the  month  from  date  of  the  paper ;  and  from 
this  result,  continue  to  subtract  the  number  of  days 
contained  in  the  months  following,  until  the  remain- 
der is  less  than  30  (except  in  case  of  February),  to 
which  add  three  days  of  grace,  and  you  will  have  the 
date  of  maturity. 
13 


i[46      ORTON  &  Sadler's  calculator. 

Example. — Note,  dated  March  10,  1874.  Payable 
ninety  days  after  date.     Find  the  date  of  maturity. 

Time,  days 90 

Number  of  days  from  March  10  to  April  1 . .  21 

"69 
April 30 

"39 
May , 31 

~8 
Add  days  of  grace 3 

Date  of  maturity June  11 

PROOF. 

Days  remaining  in  March. .  21 

"   April  ..  30 

"  "  May....  31 

"  "  "   June...  11 

93  ds.,  including  3  ds.  gr. 

Note. — "When  a  Note  or  Draft  falls  due  on  Sunday 
or  any  legal  holiday  authorized  by  the  State  or  Gen- 
eral Government,  it  must  be  paid  on  the  day  pre- 
vious. Should  a  legal  holiday  occur  on  Monday,  all 
paper  maturing  on  that  day  must  be  paid  on  the 
Saturday  previous. 

When  the  time  of  a  Note  or  Bill  is  given  in  days, 
the  days  of  date  and  maturity  are  cownted  hut  one. 

Commercial  Paper  falling  due  on  the  30th  or  3l8t 
of  any  month  which  contains  only  28,  29,  or  30  days, 
becomes  due  on  the  last  day  of  the  month,  hence  is 
legally  due  on  the  3d  of  the  month  following. 


STERLING  EXCHANGE. 


Sterling  Exchange  consists  of  Bills,  prin- 
cipally issued  by  Banks  and  Bankers  upon  their 
correspondents  in  different  countries,  to  be  used 
in  settlement  of  balances,  and  conducting  of 
business,  without  the  necessity  and  risk  of  special 
gold  remittances. 

PREMIUM  AND  DISCOUNT, 

On  Bills  of  Exchange,  is  regulated  by  the  sup- 
ply and  demand,  the  same  as  any  marketable 
commodity. 

They  command  a  premium  when  the  balance 
of  trade  is  against  the  country  where  issued,  and 
are  subject  to  discount  when  in  its  favor. 
147 


148      orton's  lightning  calculator. 

bills  of  exchange 
Are  drawn  in  sterling  money,  the  denominations 
of  which  are  shown  as  follows  : 

4  farthings  equal  1  penny — d.  I    2  shillings  equal  1  florin — fl. 
12  pence   equal   1  sLilli.ig — s.  |  20  shillings      "      1  pound — £. 

The  reduction  or  value  of  English  money  in 
U.  S.  Gold  coin  is  now  based  upon  the  IT.  S. 
Standard  Value  of  $4.8665  to  the  pound,  in  ac- 
cordance with  the  new  Act  of  Congress,  which 
went  into  effect  January  1,  1874. 

This  is  equal  to  9i%  premium  on  the  old  par 
value  of  $4.44|  to  the  £.  The  following  invalu- 
able tables,  for  the  use  of  bankers  and  business 
men,  will  save  much  time  and  labor  in  exchange 
calculations.    See  pages  152-153. 

We  also  present  the  old  method  of  calculations. 

FOREIGN  EXCHANGE  QUOTATIONS 

Are  commonly  based  upon  the  nominal  value  of 

the  £  sterling,  which  is  $4.44  J. 

The  true  value  of  the  £  sterling  is. ..$4.8675 
Nomiual      ''         "  "  "   ...  4.4444— 


Difiference 4230 

Amounting  to  nearly  9  J  of  the  nominal.  There- 
fore, when  the  £  is  quoted  at  109 J,  it  is  really 
at  par. 

STERLING  EXCHANGE  TO  U.  S.  MONEY. 

£  —  s — d  —  reduced  to  Dollars  and    Cents, 
Rule. — Reduce  the  pounds,  shillings,  and  pence  to 


STERLING    EXCHANGE.  149 

sixpences,  and  divide  by  9,  the  quotient  will  he  the 
result  in  dollars  and  cents  ;  and  for  every  'penny  ex- 
ceeding six  in  the  given  number  of  pence,  add  two 
cents  additional. 

Note. — Whenever  Exchange  commands  a  premium, 
the  per  cent,  of  premium  must  be  added  to  ascertain 
the  true  value ;  when  at  a  discount,  the  per  cent, 
must  be  deducted. 

.  Explanation  of  Rule. —  The  nominal  value  of  Hit 

pound  sterling  in  U.  S.  money  being  ($4.44|)  $4J  = 

Y  dollars;  and  in  English  money ^  20  shillings,  or 

40  sixpences.     Therefore — 

40     sixpences     =    Y     dollars. 

1  "  =     i 

9  **  =      1  " 

Hence,  the  division  by  9  according  to  the  Rule. 

PROCESS    OF    REDUCTION. 

£ — s — d — to  Sixpences. 

Rule. — Multiply  the  pounds  by  40,  and  the  shillings 
by  2  ;  to  the  product  add  1  whenever  the  pence  equal 
or  are  in  excess  of  6. 

ILLUSTRATION. 

Example  I. — Reduce  £130  9s.  Sd.  to  sixpences. 

£  130     X     40    =    5200    sixpences, 
s.      9     X       2     =        18 

d.      8    -T-      6    = 1 

5219 


150      ORTON  &  Sadler's  calculator. 

Example  II. — What  is  tlie  par  value  of  £112  9s, 
lid.,  in  U.  S.  gold  ? 

Process.  —  £  112  x  40  ^  4480  sixpences, 
s.       9   X     2  =       18 
d.    11  -^    6  -=        1 

9 ) 4499 

499.89 
Excess  of  pence  5  X  2  —  .10 

$499.99  U.  S.  gold. 

EXCHANGE,  WHEN  AT  A  PREMIUM,  OR  ABOVE  PAR. 

Example  III.  —  What  is  the  value,  in  U.  S.  gold, 
of  £162  10s.  9d.,  premium  10^  per  cent.  ? 

Process.  — £  162   X  40  =  6480  sixpences, 
s.     10  X     2  ==      20 
d.      9-^-6  =- 1 

9 )  650l  ^* 

$722.33 
Excess  of  pence  3x2  =  6 

$722.39 
Premium  10^  per  cent.       1.105 

361195 
72239 
72239 


$798.24095 

$798.24  U.  S.  gold. 

STERLING    EXCHANGE  TO    U.  S.   CURRENCY, 

With  Exchange  and  Gold  at  a  premium,  or  above  par. 

Example  IV. — What  will  be  the  cost  of  a  Bill  of 
Exchange  on  London  for  £24  14s.  ? 


STERLING   EXCHANGE.  151 

Rate  of  Exchange 1.07| 

Gold l.OsI 

Process.  —  £  24  X  40  =  960 
s.  14  X    2  =    28 

97988 

109.777  gold  =  109.78 
Exchange  quoted  1.07f 1 .0775 

54890 
76846 
76846 
10978 


Cost  in  gold $118.28 

Gold  quoted  1.08i 1.0875 

59140 
827^6 
94624 
11828 


Cost  in  U.  S.  currency $128.63 


TABLE    OF    STERLING    EXCHANGE, 
Showing  the  value  of  1  pound  to  1000  in  dollars  and  cents — calculated 


at  the 

par  value 

of  $4,444  to  £  sterling. 

£ 

$  Ct8. 

£ 

$  Ct8. 

Sh, 

$  Ct8, 

1 

4.444 

20 

88.889 

1 

.222 

2 

8.889 

25 

111.111 

2 

.444 

3 

13.333 

50 

222.222 

3 

.667 

4 

17.778 

.  75 

333.333 

4 

.889 

5 

22.222 

100 

444.444 

5 

1.111 

6 

26.G67 

200 

888.889 

6 

1.333 

7 

31.111 

250 

1111.111 

7 

1.778 

8 

35.556 

500 

2222.222 

10 

2.222 

9 

40.000 

750 

3333.333 

15 

3.333 

10 

44.444 

1000 

4444.444 

20 

4.444 

ENGLISH 


TABLE    FOR   THE 

MONEY    TO 


1    IPOTJISriD      =      $4.8S65 


0 

1 

2 

3 

4 

1 

4.8665 

53.5315 

58.3980 

63.2645 

68.1310 

2 

9.7330 

102.1965 

107,0630 

111.9295 

116.7960 

3 

14.5995 

150.8615 

155.7280 

160.5945 

165.4610 

4 

19.4660 

199.5265 

204.3930 

209.2595 

214.1260 

5 

24.3325 

248.1915 

253.0580 

257.9245 

262.7910 

6 

29.1990 

296.8565 

301.7230 

306.5895 

311.4560 

7 

34.0655 

345.5215 

350.3880 

355.2545 

360.1210 

8 

38.9320 

394.1865 

399.0530 

403.9195 

408.7860 

9 

43.7985 

442.8515 

447.7180 

452.5845 

457.4510 

Is.  Equals  24  133-400  Cts.    Id.  Equals  2  133-4800  Cts. 


O      1  I  2  I   3 


5       6 


.0202 
.0405: 
.0608; 
1.0811 1 
1.1013 

„      i.1216: 

7  .1419 

8  i'.1622 

9  '.1824i 

10  1.2327; 

11  1.2230' 


.2433  .4866 
,2636 '.5069 1 
.2838  .5272' 
,3041  .5474; 
,32441.5677; 
,3447  .5880 1 
,3649l.60S3| 
,38521.6285' 
,40551.6488; 
,42581.66911, 
,44601.68941 
4663|.709'j| 


,72991  , 
,75021  . 
,7705  {I, 
7908  ;1, 
811011, 
,831311 
8510  1. 
8719: I 
8921:1, 
9124;  1, 
,9327:1, 
9530 1 1, 


.973311 
.9935  1 
.0138  il 
.0341 !  1 
.0544  1. 
.0746!! 
.094911 
1152 I 1 


1356 
1557 
1760 
1963 


.2166  1 
.236911 
.257111 
.277411, 
,2977  il 
.3180 1 1, 
.3382 il 
,358511, 
,378811, 
.3991  1 
.4193  1 
.4396  ;i 


,4599  1.7032 
,4802  1.72^5 
.5005  1.7438 
5207  1.7641 
,5410  1  7843 
561 3 1 1.8046 
.581611.8249 
.6018 1 1^152 
.6221'!  8654 
.6424il.88;)7 
.6627l.e0u9 
.6829  1.9263 


8       9 

1.946612.1899 
1.9668:2.2102 
1.987112.2.304 
2.0074  2.2  07 


2.0277 
2.0479 
2.0682 
2.0885 
2.1088 
2.1290 
2.1493 
2.1 69C 


2.2710 
2.2913 
2.3115 
2.3318 
2.3521 
2.3724 
2.3926 
2.4129 


Note. — To  find  the  value  of  any  number  of  pounds 
represented  by  one  figure,  find  the  figure  in  the  left-hand 
margin  of  the  upper  table,  and  its  value  will  appear  in  the 
column  adjoining,  opposite  that  figure.  To  find  the  value 
when  expressed  by  two  figures,  look  for  the  tens  in  the 
left-hand  column, *and  for  the  units  in  the  top  margin, 
and  the  value  will  be  shown  in  the  ])lace  where  the  two 
columns  meet;  thus,  the  value  of  £39  is  $189.7935.  To 
find  the  value  of  £252,  look  for  25  as  before,  and  move 
the  decimal  point  one  place  to  the  right,  and  it  shows 
$1216.625;  then  add  £2  as  already  shown,  $9.7330,  and  it 
gives  the  sum  of  $1226.358. 

152 


REDUCTION    OF 

U.    S.    GOLD   COIN. 


—  iD^r  TJ.  s.  GtOXjId. 


72.9975 
121.6625 
170.3275 
218.9925 
267.6575 
316.3225 
364.9875 
413.6525 
462.3175 


6 

7 

77.8640 

82.731)5 

126.5290 

131.3955 

175.1940 

180.0605 

223.8590 

228.7255 

272.5240 

277.3905 

321.1890 

326.0555 

369.8540 

374.7205 

418.5190 

423.3855 

467.1840 

472.0505 

8 

9 

87.5970 

92.4635 

136.2620 

141.1285 

184.9270 

189.7935 

233.5920 

238.4585 

282.2570 

287.1235 

330.9220 

335.7885 

379.5870 

384.4535 

428.2520 

433.1185 

476.9170 

481.7835 

1  far.  Equals  $  .00506. 


10     11 


2.4332  2.G760 
2.4535 '2.69ns 
2.4738 '2.7 171 
2.4940 '2.7374 
2.51-131 2.7576 
2.534612.7779 
2.5549 1 2.7982 
2.5751  2.8185 
2.5954  2.8387 
2.6157  2.8590 
2.6360  2.8793 
2.a562' 2.8996 


12  I  13 

3.1632 
3.18;i5 
3.2037 
3.2240 
3.2443 
3.2646 
3.2848 
3.3051 
3.3254 
3.3457 
3.3659 
3.3862 


2.9190 
2.9401 
2.9()04 
2.9807 
3.0(110 
3.0212 
3.0415 
3.0618 
30821 
3.1023 
3.1226 
3.1429 


14 


3.4065 
3.4268 
3.4471: 
3.4673 
3.4876' 
3.5079; 
3.5282 
3.5484 
3.5687; 
3.5890 1 
3.60931 
3.62951 


15 


3.6498  {3, 
3.6701  3 
3.0904:3 
3.7107;  3, 
3.730913, 
3.751213, 
3.7715  4, 
3.7918 
3.8120 
3.8323 
3.8526  ., 
3.8729 1 4, 


16  I  17  18 

.8932;  4.1365 '4.3798 
.9134'4.1568'4.4001 
.933714  1770  4.4204 
.95401 4.1973!  4.4406 
.97431 4.2176  J4.4C09 
.9945 14.2379 1 4.48 12 
.014814.2581  4.5015; 
,035114.278414.5217! 
,0554  4.2987 14.5420! 
,0756 '4.3 190 1 4.5623: 
.0959,4.3392.4.5826; 
11 62 1 4.3595 1 4.6028 1 


The  lower  table  sho^s  the  value  of  every  combination 
of  shillings  and  pence  les.s  than  £1 ;  the  upper  margin 
representing  the  shillings,  and  the  left-hand  margin  the 
pence.  Thus,  to  find  the  value  of  13  shillings  and  6  pence, 
follow  the  column  13  downward  until  it  meets  the  left-hana 
column  opposite  6,  and  it  shows  $3.28.  By  this  method, 
any  number  of  pounds,  shillings,  and  pence  can  be  re- 
duced to  Uniteil  States  gold  quickly  and  accurately. 

To  ascertain  the  value  of  £,  s.,  and  d.  in  U.  S.  currency, 
multiply  the  amount  after  the  reduction,  per  the  above 
tables,  by  $1.00,  added  to  the  current  rate  of  premium  on 
gold. 

153 


MAKKING    GOODS. 

In  marking  goods 
it  is  the  custom  with 
most  business  houses 
to  use  a  private  mark 
to  denote  the  cost 
and  selling  price  of 
the  different  articles. 
Various  devices  are 
used  to  render  the 
cost  and  selling  price 
marks  from  being  understood  by  any  except  to 
those  employed  in  the  establishment,  whose  duty 
it  is  to  exhibit  and  sell  goods. 

Any  word  or  phrase  containing  ten  different 
letters   or   characters   is   selected,  and   used   to 
represent  the  nine  digits  and  cipher. 
To  illustrate  we  will  take  the  followino: : 


GOD 
128 


HELP 

4567 


US. 


X. 

0 


It  is  required  to  write  a  tag  or  article  showing 
the  cost  and  selling  price. 

Assuming  the  cost  to  be  S2.75,  and  selling 
price  $3.68,  the  proper  mark  will  be  ope-dlu. 

An  extra  letter  called  a  repeater  is  used  to 
prevent  the  repetition  of  a  letter  or  figure. 
154 


MARKING    GOODS.  155 

Thus,  instead  of  writing  755  according  to  the 
key  word,  which  would  be  hee,  the  repeater  K  or 
any  other  letter  not  shown  in  the  key  could  be 
used,  which  would  make  755  read  heL  The 
object  of  the  repeater  is  to  prevent  any  cue  being 
given  to  the  private  mark.  Fractions  may  be 
used  with  the  letters  or  characters  same  as 
figures.  It  is  usual  to  write  the  cost  mark  above 
the  line,  and  the  selling  price  below,  or  vice  versa, 
thus,  -21'  The  rate  of  discount  from  the  long 
price,  in  favor  of  the  wholesale  purchaser,  is 
sometimes  written  as  shown  in  the  above  illus- 
tration. 

Instead  of  letters,  characters  or  signs  may  be 
used.  The  following  words  or  phrases  are 
adapted  for  key  words  : 

BLACKHORSE.  IMPORTANCE. 

CASH  PROFIT.  NOW  BE  SHARP. 

ASKING  PRICE  AND  DISCOUNTS. 
It  is  customary  among  jobbers  and  wholesale 
dealers,  in  selling  their  merchandise  or  wares,  to 
allow  certain  discounts  from  the  trade  or  asking 
price.  In  this  connection  we  desire  to  call  atten- 
tion to  the  fact  that  losses  may  arise  when  sup- 
posed profits  are  being  made. 

TO  ILLUSTRATE. 
Suppose  our  asking  price  is  30%  above  the 


156      ORTON  &  Sadler's  calculator. 

cost,  and  we  offer  a  wholesale  discount  of  15%, 
our  profit  is  not  15%  but  101%  ;  for  the  reason 
the  discount  is  7iot  calculated  on  the  cod  but 
upon  the  asking  price,  which  includes  the 
original  cost  and  the  per  cent,  of  profit  added. 

Example. — Sold  an  invoice  of  goods  which 
cost  us  $100;  our  asking  price  being  30%  ad- 
vance or  $130,  from  which  we  allowed  a  whole- 
sale discount  of  15% — what  was  the  real  per 
cent,  of  gain?  Ans,  10^%. 
PROCESS. 
100  ^0—  Cost,  or  $  100. 

30  %— Amount  added  for  profit,   "         30. 

130  (fo—  Asking  price,  "       130. 

15%  discount  on  $130=19i%. 
19 J%—    Wholesale  discount,      or         19-50 


110i%=    Wholesale  net  price,      "        110.50 
100  %^  Cost,  "        100. 


10^%—  Net  profit,  "       $10.50^7is. 

Example  II. — Sold  an  invoice  of  hats  cost- 
ing $100,  they  having  been  marked  to  sell  at  a 
profit  of  40%,  but  in  consideration  of  cash  pay- 
ment we  allow  a  special  discount  amounting  to 
30%  fron_  our  asking  price — what  is  our  net 
remit  f    Ans.  A  loss  of  2%. 

PROCESS. 

1009^0^  Cost,  or  $100. 

40%^^    Amount  added  for  profit,      " 

140%^^  Asking  price,  ** 

42%— 30%  from  asking  price,  $140,  " 
98%t=r       Amount  of  cash  sales,       " 

100%-  Cost, 

^%%-=--  Amount  received, 

2%-=  Net  loss, 


MARKING   GOODS.  157 

TO   MARK    GOODS 

That  a  discount  may  be  allowed  from  the  asking 

price  and  have  them  net  the  desired  profit. 

Rule. — Assume  100  to  be  the  new  or  fancy 

price,  from  which  deduct  the  proposed  discount; 

then  ascertain  how  many  times  the  amount  is  con- 

tained  in  the  net  or  original  price  asked. 

Note. — The  amount  will  generally  represent  hundredths 
(100),  but  to  avoid  fractions  it  may  be  extended  to  thou- 
sandths (1000),  or  tens  of  thousands  (10,000),  in  which 
case  it  will  only  be  necessary  to  add  to  the  original  asking 
price  two,  three,  or  four  cii)hers,  as  the  case  may  require. 

Example  I. — A  manufacturer  wishes  to  offer 
the  trade  a  line  of  goods,  which  he  desires  to 
net  $17.00  per  dozen — what  must  be  his  asking 
price  to  enable  him  to  allow  a  discount  of  15%? 
Ans.  S20  per  dozen. 

PROCESS. 
Net  price,  $17.-^85=820. 
Solution, — Asking  price  assumed,  100 
Less  proposed  discount,    15 

"85)1700:820. 

Example  II.— A  wholesale  and  retail  mer- 
chant wishes  to  mark  his  goods  and  wares  at 
such  a  price  as  to  enable  him  to  offer  them  to 
his  wholesale  customers  at  a  discount  of  25%, 
and  still  gain  15%  above  first  cost — what  must 
be  his  asking  or  retail  price  for  goods  costing  $12? 
14 


158      ORTON  &  Sadler's  calculator. 

PROCESS. 

Cost,  12.00+1.80^13.80—75=^18.40  Retail  price. 

Solution,  Proof. 

Cost,                             12.00  18.40  Retail  price. 

Wholesale  price  15%  1.80  4.60  25%  discount. 

Asking  price,             j^80  13.80  Wholes'e  price. 

Retail  price  assumed,   100  1.80  15%prof.deduc. 

Less  proposed  disc't,      25  $12^0  Cost 
"75)138000(18.40 
_75_ 

630 
600 

300 
300 


DISCOUNT  FEOM   ASKING   PRICE. 

To  ascertain  the  per  cent,  of  discount  from  the 
asking  price  to  produce  the  cost. 

Rule. — From  the  asJcing  price  subtract  the  cost, 
divide  the  difference  by  the  asking  price,  and  the 
quotient  will  be  the  per  cent,  of  discount. 

Example. — The  asking  price  of  a  case  of 
goods  costing  S120  was  $150 — what  per  cent, 
of  discount  can  be  offered  to  have  the  goods  net 
cost?    Ans.  20%. 

Sohdion.  Proof. 

$150  Asking  price.  Asking  price,        $150 

120  Cost.  Discount  20%=  __30 

~30-r-150=20%.  Cost,  $120 


RAPID  PROCESS  OF  MARKING  GOODS. 

4  VALUABLE  HINT  TO  MERCHANTS  ANI    ALL  RETAIL  DEALERS 
IN  FOREIGN  AND  DOMESTIC  DRY   GOODS. 

Retail  merchants,  in  buying  goods  by  whole- 
sale, buy  a  great  many  articles  by  the  dozen,  such 
as  boots  and  shoes,  hats  and  caps,  and  notions  of 
various  kinds.  Now,  the  merchant,  in  buying,  for 
instance,  a  dozen  hats,  knows  exactly  what  one  of 
those  hats  will  retail  for  in  the  market  where  he 
deals ;  and,  unless  he  is  a  good  accountant,  it  will 
often  take  him  some  time  to  determine  whether  he 
can  afford  to  purchase  the  dozen  hats  and  make  a 
living  profit  in  selling  them  by  the  single  hat ;  and 
in  buying  his  goods  by  auction,  as  the  merchant 
often  does,  he  has  not  time  to  make  the  calculation 
before  the  goods  are  cried  off.  He  therefore  loses 
the  chance  of  making  good  bargains  by  being 
afraid  to  bid  at  random,  or  if  he  bids,  and  the 
goods  are  cried  off,  he  may  have  made  a  poor  bar- 
gain, by  bidding  thus  at  a  venture.  It  then  be- 
comes a  useful  and  practical  problem  to  determine 
instantly  what  per  cent,  he  would  gain  if  he  re- 
tailed the  hats  at  a  certain  price. 

To  tell  what  an  article  should  retail  for  to 
make  a  profit  of  20  per  cent., 

Rule. — Divide  what  the  articles  cost  per  dozen  by 
10,  which  is  done  hy  removing  the  decimal  point  om 
place  to  the  left. 

159 


160      ORTON  &  Sadler's  calculator. 

For  instance,  if  tats  cost  $17.50  per  dozen,  re 
move  the  decimal  point  one  place  to  the  left,  mak- 
ing $1.75,  what  they  should  be  sold  for  apiece  to 
gain  20  per  cent,  on  the  cost.  If  they  cost  $31.00 
per  dozen,  they  should  be  sold  at  $3.10  apiece,  etc. 
We  take  20  per  cent,  as  the  basis  for  the  following 
reasons,  viz. :  because  we  can  determine  instantly, 
by  simply  removing  the  decimal  point,  without 
changing  a  figure;  and,  if  the  goods  would  not 
bring  at  least  20  per  cent,  profit  in  the  home  mar- 
ket, the  merchant  could  not  aflford  to  purchase  and 
would  look  for  goods  at  lower  figures. 

Reason. — The  reason  for  the  above  rule  is  ob- 
vious :  For  if  we  divide  the  cost  of  a  dozen  by  12, 
we  have  the  cost  of  a  single  article ;  then  if  we 
wish  to  make  20  per  cent,  on  the  cost,  (cost  being 
1  ^^  T'))  ^®  ^^^  ^^®  2^  P^^  cent.,  which  is  ^,  to 
the  ^,  making  f  or  ||^ ;  then  as  we  multiply  the 
cost,  divided  by  12,  by  the  i|  to  find  at  what  price 
one  must  be  sold  to  gain  20  per  cent.,  it  is  evident 
that  the  12s  will  cancel,  and  leave  the  cost  of  a 
dozen  to  be  divided  by  10,  which  is  done  by  re- 
moving the  decimal  point  one  place  to  the  left. 

1.  If  I  buy  2  doz.  caps  at  $7.50  per  doz.,  what 
shall  I  retail  them  at  to  make  20^?     Ans.  75  cts. 

2.  When  a  merchant  retails  a  vest  at  $4.50  and 
makes  20^,  what  did  he  pay  per  doz.?    Ans.  $45 

3.  At  what  price  should  I  retail  a  pair  cf  boota 
that  cost  $85  per  doz.,  to  make  20%?    Ans.  $8.50. 


MARKING   GOODS.  161 

RAPID  PROCESS  OF  MARKING  GOODS  AT  DIFFERENT 

PER  CENTS. 

Now,  as  removing  the  decimal  point  one  place 
to  the  left,  on  the  cost  of  a  dozen  articles,  givoF 
the  selling  price  of  a  single  one  with  20  per  cent, 
added  to  the  cost,  and,  as  the  cost  of  any  article 
is  100  per  cent.,  it  is  obvious  that  the  selling  price 
would  be  20  per  cent,  more,  or  120  per  cent.; 
hence,  to  find  50  per  cent,  profit,  which  would 
make  the  selling  price  150  per  cent.,  we  would 
first  find  120  per  cent.,  then  add  30  per  cent.,  by 
increasing  it  one-fourth  itself;  to  make  40  per 
?ent.,  add  20  per  cent.,  by  increasing  it  one-sixth 
itself;  for  35  per  cent.,  increase  it  one-eighth  itself, 
etc.  Hence,  to  mark  an  article  at  any  per  cent, 
profit,  we  have  the  following 

GENERAL  RULE 

First  find  20  per  cent,  profit^  hy  removing  the 
decimal  point  one  place  to  the  left  on  the  price  the 
articles  cost  a  dozen;  then,  as  20  per  cent,  profit  is 
120  per  cent.,  add  to  or  subtract  from  this  amount 
the  fractional  part  that  the  required  per  cent,  added 
to  100  is  more  or  less  than  120. 

Merchants,  in  marking  goods,  generally  take  a 
per  cf»nt.  that  is  an  aliqot  part  of  100,  as  25%. 
i{3^%,  50%,  etc.  The  reason  they  do  this  is  be- 
cause it  makes  it  much  easier  to  add  such  a  per 
y^At,  to  the  cost;  for  instance,  a  merchant  could 


162   ORTON  &  Sadler's  calculator. 

mark  almost  a  dozen  articles  at  50  per  cent,  profit 
in  the  time  it  would  take  him  to  mark  a  single 
one  at  49  per  cent.  For  the  benefit  of  the  student, 
and  for  the  convenience  of  business  men  in  mark- 
ing goods,  we  have  arranged  the  following  table : 

TABLE 

For  Marking  all  Articles  hmight  hy  the  Dozen. 
N   B.  Most  of  these  are  used  in  business. 
To  make  20  fo  remove  the  point  one  place  to  the  left 

K  U  8Q^  U  U 

"      "      44%       "  " 


37^% 
35/^ 
3345 
32  f^ 
30^ 
28  5 
2&fc 
25  f^ 

m% 

16ff. 


and  add   J 

itself. 

u 

u 

(i 

u 

" 

a 

u 

u 

u 

u 

t( 

u 

u 

u 

u 

u 

i( 

u 

u 

(( 

(t 

u 

(( 

A 

t( 

u 

(( 

A 

(C 

(( 

u 

A 

u 

(( 

u 

A 

u 

(( 

u 

A 

u 

subtract 

A 

(( 

u 

A 

u 

If  T  buy  1  doz.  shirts  for  $28.00,  what  shall  1 
retail  them  for  to  make  50%?  Ans.  83.50, 

Explanation. — Remove  the  point  one  place  tc 
th«*  left,  and  add  on  |  itself. 


BASIS  OF  SUCCESS  IN  BUSINESS. 

SHORT    CREDITS    SECURE    LARGE    PROFITS. 

Be  watchful  of  your  credits,  as  they  have  much  to  ilo 
toward  securing  large  profits  iu  business. 

The  strong  argument  in  favor  of  such  a  course  can  best 
be  illustrated  from  the  results  shown  in  the  following 
table,  giving  the  accumulations  from  $100,  invested  in 
business  for  the  term  of  ten  years,  and  turned  over  tit 
profits  of  5,  8,  and  10%. 

TABLE. 
Capital  invested.  ^"^^°*'     ^"^g^^^"*'     ^"1^:!"^' 

$100  turned  over  every  3  mos.    703.98     2172.46     4525.9H 
100        "         "         "       6     "        265.33       466.08       672.74 
100        "        ''        "     12    "        162.87       215.88       259.3() 
100        "        "        "      2  years.  127.62       146.93       161.05 
Note. — An  important  question  to  be  considered  in  re- 
ducing prices  is,  can  one's  sales  be  sufiiciently  increased, 
so  as  to  compensate  for  the  reduction  of  profits?     From 
the  above  table  it  will  be  observed  that  quick  sales  and 

SMALL  PROFITS 
Are  more  desirable  than  large  ones,  when  secured  at  the 
expense  of  long  credits. 

BEWARE  OF  EXPENSE. 

As  expenses  diminish  all  profits  it  is  essential  that  judi- 
cious economy  be  at  all  times  exercised.  Man^  a  business 
becomes  bankrupt  from  lack  of  foresight  in  this  one 
particular. 

LOOK  TO  YOUR  CREDIT. 

Always  command  a  thorough  knowledge  of  your  busi- 
ness and  your  books. 

A  smaller  business  with  cash  capital  produces  larger 
profits  than  a  large  business  conducted  on  credit. 

Be  not  too  anxious  to  extend  your  business  or  branch  out. 

Have  a  smaller  house  and  larger  capital. 

Goods  well  bought  and  discounts  saved  secure  early 
profits. 

Avoid  outside  speculation,  the  chances  are  against 
success. 

Do  not  overtrade :  goods  in  stock  are  better  than  charged 
up  iu  bad  debts. 

163 


Ledger  accounts  arise  from  business  transac- 
tions, and  comprise  a  series  of  condensed  state- 
ments under  specific  titles,  showing  one's  relation 
with  persons  and  property. 

The  left-hand  side  of  each  account  shows  debit 
entries,  and  the  right-hand  side  credit  entries. 

CLASSIFICATION   OF   ACCOUNTS. 

Accounts  are  divided  or  classified  under  two 
distinct  heads,  i,  e.,  Beal  and  Representative. 

Real  Accounts  are  those  exhibiting  Resources 
or  Liabilities. 

Representative  Accounts  are  those  exhibiting 
Gains  or  Losses. 

The  several  styles  of  accounts,  such  as  are 
used   in  the  most  extensive   business  establish- 
ments, g.re  herewith  presented,  showing  in  detail 
the  office  or  exhibit  of  each.     To  the 
164 


LEDGER    ACCOUNTS.  165 

STUDENT  OR  ACCOUNTANT, 
Who   is  desirous  iu  posting  himself  regarding 
the  priuciples  governing  accounts,  and  bahmc- 
ing  or 

CLOSING  THE  LEDGER, 
The  following  work  or  illustrations  will  prove 
invaluable,  and  will  amply  repay  for  any  time 
or  labor  expended,  as  it  contains  in  a  condensed 
form  all  that  is  of  practical  value  appertaining 
to  the  subject. 


/ 


ccoan. 


Represents  the  proprietor,  showing  his  capital  or 
investment, 

AT    COMMENCING   BUSINESS. 

The  Credit  of  Account  shows  the  Capital  or 
Resources  invested. 

The  Debit  of  Account  shows  the  Liabilities  or 
Amounts  assumed  to  be  paid  from  the  business. 

The  Difference  shows  the  Net  Capital,  if  the 
credit  is  in  excess ;  and  the  Net  Insolvency,  if  the 
debit  is  iu  excess. 

DURING    BUSINESS. 

The  Credit  of  Account  will  show  any  additional 
investments. 


166      ORTON  &  Sadler's  calculator. 

The  Debit  of  Account  will  show  any  additional 
liabilities  assumed,  or  withdrawals  of  capital. 

UPON   CLOSING  THE   LEDGER. 

The  Credit  of  Account  will  show  the  total  in- 
vestment of  capital  and  the  Net  Gain  arising 
from  the  business. 

The  Debit  of  Account  will  show  the  total 
liabilities  assumed,  withdrawals  of  capital,  and 
Net  Loss  arising  from  the  business. 

The  Difference  will  show  the  proprietor's 
Present  Worth  or  Insolvency.  If  the  credit  side 
is  in  excess,  a  Present  Worth ;  if  the  debit  side  is 
in  excess,  an  Insolvency. 

The  account  is  closed  To  or  By  Balance,  as 
shown  by  the  excess. 

Are  treated  precisely  like  the  stock  or  capital 
account.  The  Net  Interests  of  all  the  partners 
taken  together  will  exhibit  the  condition  or  stand- 
ing of  the  firm. 


Classification,  Real ;  Closed  By  Balance. 


LEDGER    ACCOUNTS.  167 

Debit  of  Account  shows  Total  Cash  Receipts. 

Credit  of  Account  shows  Total  Cash  Payments. 

The  Difference  shows  a  Resource  consisting  of 

the  amount  of  cash  on  hand. 

Note. — As  more  cash  cannot  be  paid  out  than  is  re- 
ceived, the  credit  of  the  account  can  neier  he  iti  excess. 


Classification,  Real ;  Closed  B\j  Balance. 

Debit  of  Account  shows  others'  notes  and  ac- 
ceptances received. 

Credit  of  Account  shows  others'  notes  and 
acceptances  paid,  discounted  or  disposed  of. 

The  Difference  shows   a   Resource  consisting 

of  others^  paper  on  hand,  the  value  of  which  we 

are  to  receive. 

Note  — As  others'  paper  cannot  be  disp'jsed  of  until 
received,  the  cre'Jit  of  the  account  can  never  be  in  excess. 


Classification,  Real ;  Closed  To  Balance. 

Credit  of  Account  shows  our  notes  and  accept- 
ances issued. 

Debit  of  Account  shows  our  notes  and  accept- 
ances that  have  been  paid  or  redeemed. 


168      ORTON  &  Sadler's  calculator. 

The  Difference  shows  a  Liability  consisting  of 

our  outstanding  or  unredeemed  paper. 

Note. — As  we  cannot  redeem  or  pay  more  of  our  paper 
than  is  issued,  the  debit  side  of  the  account  can  never  be 


in  excess. 


lU. 


eUonwv    (<^W-(>cow7i\ 


Classification,  Real ;  Closed  To  or  By  Balance, 

Debit  oj  Account  shows  our  charges  against 
them  or  indebtedness  in  our  favor. 

Credit  of  Account  shows  their  charges  against 
us,  or  our  indebtedness  in  favor  of  others. 

The  Difference,  if  in  favor  of  the  debit  side, 
shows  a  Resource,  or  the  sum  due  us  from  the 
parties  represented  by  the  account. 

If  in  favor  of  the  credit  side,  a  Liability,  or 

amount  we  owe  them. 

Note. — When  the  balance  is  in  our  favor  the  account 
is  termed  a  Personal  Account  Receivable;  when  against 
us  a  Personal  Account  Payable. 


u. 


ond'tanryien^ 


Accounts  representing  consignments  of  mer- 
chandise or  property  received  from  others,  to  be 
sold  for  their  account  and  risk,  are  treated  the 
same  as  Personal  Accounts.     The  debit  side  of 


LEDGER    ACCOUNTS.  169 

the   account  showing  amount   of  expenses   in- 
curred, and  the  credit  side  the  returns. 


Q/iiett>fhwriwu^. 


Classification,  Kepresentative ;  Closed  To  or 
By  Loss  and  Gain, 

Debit  of  Account  shows  Cost  of  goods  pur- 
chased. 

Credit  of  Account  shows  Sales  or  Proceeds. 

Note. — If  the  goods  are  not  all  sold  or  disposed  of,  it 
will  be  necessary  to  credit  the  account  with  the  inventory 
or  market  value  of  the  unsold  quantity. 

The  Difference  shows  a  gain  or  loss  according 

to  the  excess  of  the  sides.     If  in  favor  of  the 

credit,  Si  Gain;  of  the  debit,  ^  Loss. 

Note. — This  account  may  be  made  to  comprise  all 
properties  purchased  for  traffic,  such  as  groceries,  dry- 
goods,  flour,  produce,  hardware,  crockery,  etc. ;  or,  if  de- 
sired, each  kind  may  be  represented  under  its  own  separate 
heading. 


^^m^7 


fun/menu  ¥(> 

A  special  account  under  the  above  heading  is 
frequently  used  to  represent  merchandise  or  prop- 
erty which  we  have  consigned  away  to  be  sold 
for  our  account  and  risk.  The  account  is  treated 
similar  to  Merchandise.  It  is 
15 


170   ORTON  &  Sadler's  calculator. 

Debited  with  the  cost  or  outlay. 

Credited  with  the  returns. 

Note. — At  closing  the  Ledger,  if  account  sales  or  full 
returns  have  not  been  received,  it  will  be  necessary  to 
credit  the  account  with  the  value  represented,  or  inven- 
tory, same  as  in  Merchandise. 

The  Difference  shows  a   Gain  or  Loss,     (See 
Merchandise  account.) 


Classification,  Representative;  Closed  To  or 
By  Loss  and  Gain. 

Debit  of  Account  shows  Cost  or  Outlay. 

Credit  of  Account  shows  Proceeds  from  Sales, 
or  Income. 

Note. — Add  inventory,  if  any,  same  as  in  Merchandise. 

The  Difference  shows  Gain  or  Loss  according 
to  the  excess  of  sides.     (See  Merchandise  ac't.) 


^^foU     Q^io^'^wted 


Classification,  Representative ;  Closed  By  Loss 
mid  Gain. 

Debit  of  Account  shows  Cost  or  Outlay. 
Credit  of  Account  shows  Returns,  if  any.     If 


LEDGER    ACCOUNTS,  171 

value  is  estimated,  add  as   Inventory  in   Mer- 
chandise. 

The  Difference  will  show  a  Loss, 

Classification,  Eepresentative  ;  Closed  By  Loss 
and  Gain. 

Debit  of  Account  shows  Expenditures  and  out- 
lay for  conducting  the  business. 

Note. — In  case  there  should  be  returns  from  expense 
expenditures,  the  account  would  be  credited. 

The  Difference  shows  a  Loss. 


n¥etm 


Classification,  Representative;  Closed  To  or 
By  Loss  and  Gain. 

Debit  of  Account  shows  sums  paid  for  use  of 
others'  money. 

Credit  of  Account  shows  sums  received  for  use 
of  our  money. 

The  Difference,  if  in  favor  of  the  credit  side, 
shows  a  gain ;  when  in  favor  of  the  debit  side,  a 
loss. 


172   ORTON  &  Sadler's  calculator. 


t^'t>ovo7^¥, 


Classification,  Representative ;  Closed  To  or 
By  Loss  and  Gain. 

Debit  of  Account  shows  sums  paid  or  allowed 
by  us. 

Credit  of  Account  shows  sums  received  or  al- 
lowed us. 

The  Difference,  if  in  favor  of  the  credit  side, 
shows  a  gain ;  when  in  favor  of  the  debit  side,  a 


nydw^wnt>&. 


Classification,  Representative;  Closed  To  or 
By  Loss  and  Gain, 

Debit  of  Account  shows  amount  paid  for  in- 
surance. 

Credit  of  Account  shows  amount  received  from 
others  for  insurance. 

The  Difference,  if  in  favor  of  the  credit  side, 
shows  a  gain ;  when  in  favor  of  the  debit  side,  a 


LEDGER    ACCOUNTS.  173' 


Classification,  Representative;  Closed  To  or 
By  Loss  and  Gain. 

Debit  of  Account  shows  suras  paid  or  allowed 
for  services  in  buying  or  selling. 

xjredit  of  Account  shows  suras  received  or  al- 
lowed for  services  in  buying  or  selling. 

Tlie  Difference,  if  in  favor  of  the  credit  side, 
shows  a  gain ;  when  in  favor  of  the  debit  side,  a 
loss, 

(or,  Frofii  and  Loss)  exhibits  Net  Gains  or 
Losses.  Closed  To  or  By  Stock  or  Capital^ 
wherein  single  proprietorship  is  represented. 
Li  Partnership  business  the  account  is  closed  To 
or  By  Partners^  Account,  showing  each  partner's 
respective  share  in  the  division  of  gains  or  losses. 

Debit  of  Account  shows  the  total  Losses  arising 
from  the  business. 

Credit' of  Account  shows  the  total  Gains  pro- 
duced from  the  business. 

The  Difference  shows  a  Net  Gain  or  Net  Loss, 
according  to  the  excess  of  sides ;  a  Net  Gain  if 


174      ORTON  &,  Sadler's  calculator. 

the  difference  is  in  favor  of  the  credit  side,  and  a 

Net  Loss  when  in  favor  of  the  debit  side. 

Note. — In  closing  the  account,  the  Net  Gain  is  carried 
to  the  credit  of  Stock  or  Partners^  account,  showing  an 
increase  of  capital.  The  Net  Loss  is  carried  to  the  debit 
of  Stock  or  Partners^  account,  showing  a  diminution  of 
capital. 

Gains  or  Losses  are  divided  between  partners 
in  accordance  with  the  articles  of  copartnership 
or  agreement  made  between  them.  The  methods 
of  adjustment  will  be  found  fully  illustrated  un- 
der the  head  of  Partnership ^  page  139  of  this 
work. 


t>ct>wrVv- 


Is  a  special  account  kept  with  the  proprietor  to 
show  his  transactions  with  the  business  as  an  in- 
dividual. 

The  account  is 

DEBITED 
With  his  withdrawals  of  money  or  the  appro- 
priation of  any  value  belonging  to  the  business 
to  his  private  use,  such  as  personal,  household, 
or  living  expenses. 

The  account  is  closed  By  Stock  or  Capital^  and 
total  amount  carried  to  the  debit  of  the  Stock  or 
Capital  account. 


CLOSING   THE   LEDGER.  175 

Are  special  accounts  kept  with  each  partner, 
showing  their  transactions  with  the  firm  as  indi- 
viduals.    The  accounts  are 

DEBITED 
For  precisely  the  same  reasons  as  explained  in 
the  individual  proprietor's  Private  Account. 

They  are  closed  By  Partner Sy  and  each  total 
debit,  as  shown,  carried  to  the  debit  of  the 
respective  partner's  general  or  Capital  Account. 


HOW  TO   CLOSE  THE   LEDGER. 

Inventories. — Take  an  inventory  of  all  unsold 
property  and  credit  the  account,  showing  the  cost 
of  same  when  purchased. 

This  entry  is  made  with  red  ink,  thus  :  By  Bal- 
ance Inventory,  or  By  Inventory.  After  ruling,  the 
amount  is  brought  down  to  the  debit  of  the  new 
account  in  black  ink. 

Representative  Accounts.  —  Those  showing 
Gains  or  Losses  are  closed  after  crediting  the 
inventory,  if  any,  by  writing  on  the  smaller  side, 
with  red  ink,  To  Loss  and  Gain,  or  By  Loss  and 


CLOSING   LEDGER   ACCOUNTS. 
176 


CLOSING   THE   LEDGER.  177 

Gain,  The  excess  as  shown  is  carried  to  the 
debit  or  credit  of  the  Loss  and  Gain  account,  the 
entry  being  made  in  black  iuk. 

Real  Accounts. — Those  showing  Resources  or 
Liabilities  are  closed  by  writing  on  the  smaller 
side,  in  red  ink,  To  Balance  or  Bfj  Balance, 

After  ruling,  the  balance  of  excess  as  shown 
is  brought  to  the  debit  or  credit  of  the  new 
account. 

Loss  and  Gain. — After  closing  all  Bepresenta- 
live  accounts,  and  the  excess  of  gains  or  losses 
having  all  been  brought  forward,  write  on  the 
smaller  side,  in  red  ink,  To  Stock  or  Partners'  or 
By  Stock  or  Partners'.  The  difference  will  show 
the  Net  Gain  or  Net  Loss,  which  carry  to  the 
debit  or  credit  of  Stock  or  Partners*  account. 

Stock  or  Partners'  Account. — Write  on  the 
smaller  side,  in  red  ink.  To  Balance  or  By  Bal- 
ance. Rule  up  the  account  and  bring  down  the 
balance  as  shown  to  the  debit  or  credit  of  the 
new  account  in  black  ink. 

Proof  of  Work. — Take  a  Trial  Balance  after 
closing,  and  if  your  work  is  correct,  the  Ledger 
will  be  in  equilibrium. 


ERKOES  IN  TRIAL  BALANCES. 

VALUABLE  RULES  FOR  THEIR  DETECTION. 

trom  Bryant  &  Stratton's  Business  Arithmetic,  with  kind  permission 

of  the  author,  11,  B.  Bryant,  President  Jiryant  &  Stratloa 

Buainetjs  College,  Chicago,  Illinois. 

In  keeping  accounts  by  Double  Entry,  each 
item  appears  in  at  least  two  different  accounts, 
on  the  Dr.  side  in  one  and  on  the  Cr.  side  in  the 
other ;  hence  the  sum  of  all  the  debit  entries  in 
all  the  accounts  will  equal  the  sum  of  all  the 
credit  entries  in  all  the  accounts,  and  the  sum 
of  the  Dr.  balances  will  equal  the  sum  of  the 
Cr,  balances,  if  all  the  entries  are  properly  made. 
178 


ERROES  IN  TRIAL  BALANCKS. 


179 


ILLUSTRATION  OF  DOUBLE    ENTRY. 
Dr.  merchandise.  Cr. 


May 


$  75 
380 
1200 
725 
1120 


$3500 


Apr.  I    1 
111  ay     10 


By  Ciisfi 


$3000 
500 


Dr. 


Al>r.  i    1     ToMdse. 
May      10 


CASH. 


$3000 
500 


Cr. 


380 
liOO 

725 
1120 


A  TRIAL  BALANCE 
Is  a  summary  of  the  entire  amounts  entered  on 
the  Dr.  and  Cr.  side  of  each  account,  or  simply 
of  the  balances  of  all  the  accounts  in  detail,  and 
if  the  sums  of  the  debit  and  credit  entries  in  the 
Trial  Balance  are  not  equal,  then  there  is  some 
error  in  the  accounts  or  in  making  up  the  Trial 
Balance,  which  should  be  discovered  and  cor- 
rected. 

The  first  rule  of  the  book-keeper  should  be  to 
make  no  error;  but  for  such  as  are  fallible  the 
following  suggestions  may  be  of  some  practical 
utility. 


180      ORTON  &  Sadler's  calculator. 

1.  If  the  error  be  found  in  one  figure  only,  it  is 
probably  an  error  of  adding  or  copying. 

2.  If  it  involve  several  figures,  it  may  have  arisen 
from  the  omission  of  an  entire  entry  or  from  making 
the  same  entry  twice. 

3.  If  it  be  divisible  by  2,  without  a  remainder,  it 
may  have  arisen  from  posting  an  item  to  the  wrong 
side  of  the  account,  in  which  case  the  item  would 
be  half  of  the  apparent  error. 

4.  If  the  error  be  divisible  by  9,  without  a  re- 
mainder, it  may  have  arisen  from  transposition, 
three  cases  of  which  may  be  easily  detected  by  rules 
founded  on  the  peculiar  property  of  the  number  9. 
These  cases  are — 

1st.  When  two  figures  are  made  to  exchange 
places  with  each  other,  the  orders  in  notation  re- 
maining the  same :  e.  g.,  372  made  to  read  327,  or 
732,  or  273. 

2d.  When  two  or  more  figures  are  made  to  change 
their  places  in  notation,  their  arrangement  in  re- 
spect to  each  other  remaining  the  same :  e.  gf., 
$4275  made  to  read  $42750,  or  $42.75,  or  $427.50. 

3d.  When  two  significant  figures  are  made  to 
change  position  both  with  respect  to  each  other 
and  also  the  orders  of  notation  :  e.  g.,  $14  made  to 
read  $0.41. 

5.  To  detect  the  first  and  second  cases  of  trans- 
position divide  the  amount  of  the  error  hi  the  trial 
balance  successively  hy  9,  99,  999,  9099,  e^a,  as  far  as 


ERBORS   IN   TRIAL   BALANCES.  181 

pofi'iible  without  a  remainder,  rejecting  all  ciphers  at 
the  right  of  the  last  significant  Jigure  in  the  error. 

The  quotients  that  contain  hut  one  digit  figure 
will  express  the  difiference  between  the  two  digit 
figures  transposed,  which  will  be  adjacent  to  each 
other  it*  the  divisor  consist  of  but  one  9,  separated 
by  one  other  figure  if  it  consist  of  two  9's,  by  two 
other  figures  if  it  consist  of  three  9*s,  and  so  an. 

'I'hose  quotients  wliich  contain  two  or  more 
figures  will  express  the  member  itself,  which  is 
transposed  in  notation  simply,  the  arrangement  of 
the  significant  figures  remaining  the  same.  In 
either  case  the  order  of  the  last  significant  figure 
in  the  error  will  be  the  lowest  order  of  the  figures 
transposed.  The  orders  of  the  other  figures  can  be 
easily  determined  by  referring  to  the  error  and 
applying  the  principles  of  notation. 

6.  To  detect  the  third  case,  divide  the  error  in 
the  balance  by  as  many  9's  as  is  passible  so  as 
to  give  only  a  single  figure  in  the  quotient,  and 
then  the  remainder  in  the  same  way,  rejecting  all 
ciphers  at  the  right  of  the  last  significant  figure 
in  both  dividends,  after  which  there  should  be  no 
remainder. 

The  first  quotient  will  be  the  figure  filling  both 
the  highest  and  lowest  order  in  the  transposition ; 
the  second  quotient  will  be  the  other  figure. 

NoiF,  —If  tlie  error  of  the  tria^  baianco  ba  not  divisible  by  9  it  caa- 
tiot  be  the  n'diilt  of  tVAnsposition  alono.  Hut  whenever  the  error 
becomes  n;>  re'hiced  m  to  be  divisible  by  9  without  a  remainder,  a 
tva'ii;)06Ui(ju  being  then  possible,  the  above  todta  boduIU  bo  a])i)lied. 

16 


182      ORTON  &  Sadler's  calculator. 

WE  ADD  THE   FOLLOWING 

IMPORTANT    SUGGESTIONS. 

First. — Examine  the  Cash  and  Bills  Receivable 
accounts  ;  the  balance  can  never  appear  on  the  credit 
side,  and  should  equal  the  amount  of  cash  and  notes 
on  hand,  as  shown  by  the  0.  B.  and  B.  B. 

Second. — Refer  to  the  Bills  Payable  account ;  the 
balance  shown  must  appear  in  favor  of  the  credit 
side  and  equal  the  amount  of  our  outstanding 
paper. 

Third. — If  the  cash  book  is  kept  as  an  original 
book  of  entry,  see  that  the  balance  on  hand  from 
previous  month  has  been  deducted. 

Fourth. — If  the  error  appears  only  in  the  cent  or 
dollar  columns,  it  is  not  necessary  to  add  the  columns 
to  the  left. 

Fifth. — If,  after  applying  the  above  tests,  the  error 
still  exists,  it  will  be  necessary  to  go  over  the  entire 
work.  Re-add  carefully  the  debit  and  credit  sides 
of  all  the  accounts,  as  the  error  undoubtedly  lies  in 
the  addition.  If  not  found,  examine  each  posting 
separately,  check  from  the  journal  or  book  posted 
from  to  the  ledger,  and  vice  versa,  as  you  proceed 
with  your  work. 

Note. — In  the  use  of  these  rules  in  practice,  not  cnJy 
the  balances  of  the  ledger  accounts  as  they  appear  on 
the  balance-sheet  should  be  examined,  but  also  all  the 
separate  postings,  as  a  transpQsitiou  there  will  equally 
affect  the  final  balance. 


MEASUREMENT  OF  LUMBER. 


The  unit  of  board  measure  is  a  square  foot  1 
inch  thick. 

To  measure  inch  hoards. 

Rule. — Multiply  the  length  of  the  board  in  feet 
by  its  breadth  in  inches,  and  divide  the  product 
by  12  ;  the  quotient  is  the  contents  in  square  feet. 

Note. — When  the  board  is  wider  at  one  end  than 
the  other,  add  the  width  of  the  two  ends  together, 
and  take  half  the  sum  for  a  mean  width. 

Example. — How  many  square  feet  in  a  hoard 
10  feet  long,  13  inches  wide  at  one  end,  a^^d  9 
inches  wide  at  the  other  ? 

Process. — (13  +  9)  -j-  2  =  11  (mean  width}  hep 
10  length  X  11  =  110  -^  12  =  9^  feet.     Arts. 
183 


184      ORTON  &  Sadler's  calculator. 

Sawed  lumber,  as  joists,  plank,  and  scantlings, 
are  now  generally  bought  and  sold  by  boai^d  meaS' 
ure.  The  dimensions  of  a  foot  of  board  measure 
are  1  foot  long,  1  foot  wide,  and  1  inch  thick. 

To  ascertain  the  contents  (board  measure)  of 
boardSy  scantling,  and  plank. 

Rule. — Multiply  the  width  in  inches  by  the 
thickness  in  inches,  and  that  produc<^  by  the  length 
Infeety  which  last  product  divide  by  12. 

Example. — How  many  feet  of  lumber  in  14 
planks  16  feet  long,  18  inches  wide  and  4  inches 
thick? 

Process.— 16  feet  X  18  inches  X  4  inches  =1152, 
then  1152  -T-  12  =  96  feet  =  contents  of  one  plank. 
96  X  14  =  1344  feet.    Ans. 

To  find  hoiv  many  feet  of  lumber  can  be  sawed 
fro'n  a  log.     (Gauge  of  saw  \  inch.) 

Rule. — From  the  diameter  of  the  log  in  inches, 
subtract  4  (for  slabs),  one-fourth  of  this  remainder 
squared  and  multiplied  by  the  length  in  feet  will 
give  the  correct  amount  of  lumber  that  can  be 
made  from  any  log  whatever. 

Example. — How  many  feet  of  lumber  can  be 
made  from  a  log  which  is  36  inches  in  diameter, 
and  10  feet  long  ? 

Process. — From  36  (diameter)  subtract  4  (f<»r 
slabs)  =  32,  then  divide  the  32  by  4,  making  8. 
which  squared  =  64,  then  multiply  the  64  by  10 
(length)  =  640  feet.    Am. 


MEASUREMENT   OF    LUMBER.  185 

To  find  how  many  fed  of  lumber  there  are  left  in  a 
log  after  it  is  made  perfectly  square. 

Rule. — Multiply  the  diameter  in  inches  at  the 
small  end  by  one-half  the  number  of  inches,  and 
this  product  by  the  length  of  the  log  in  feet,  which 
last  product  divide  by  12. 

Example. — If  the  diameter  of  a  round  stick  of 

timber  be  22  inches,  and  its  length  20  feet,  how 

much  lumber  will  it  contain  when  hewn  square? 

22  X  11  X  20 

Half  diameter  =  11,  and =  403  J  ft., 

12 

the  lumber  when  hewn  square. 

To  find  how  many  feet  of  square  edged  boards,  of  a 
given  thickness j  can  be  sawed  from  a  log  of  a  given 
diameter. 

Rule. — Find  the  quantity  of  lumber  in  the  log, 
when  made  square,  by  the  last  Rule ;  then  divide 
by  the  thickness  of  the  board,  including  the  saw 
calf,  the  quotient  is  the  number  of  feet  of  boards. 

Example. — How  many  feet  of  square  edged 
boards,  li  inch  thick,  including  the  saw  calf,  can 
be  sawn  from  a  log  20  feet  long,  and  24  inches 
diameter? 

24  X  12  X  20 

=  480  ft.,  the  lumber  when  hewn  sq. 

12 
Then  480  divided  by  It  »=  384  feet.  Ans. 


MEASUREMENT  OF  WOOD. 


Wood  is  measured  by  the  cord,  which  contains 
128  cubic  feet. 

Wood  is  bought  and  sold  by  the  cord  and  frac- 
tions of  a  cord. 

Pine  and  spruce  spars  from  10  to  4  inches  in 
liameter  inclusive,  are  measured  by  taking  the  dia- 
meter, clear  of  bark,  at  one-third  of  their  length 
from  the  large  end. 

Spars  are  usually  purchased  by  the  inch  diame- 
ter ;  all  under  4  inches  are  considered  poles. 

Spruce  spars  of  T  inches  and  less,  should  have 
5  f-^et  in  length  for  every  inch  in  diameter 
186 


MEASUREMENT   OP   WOOD.  187 

Note. — A  pile  of  wood  that  is  8  feet  long,  4  fecJl 
high,  and  4  feet  wide,  contains  128  cubic  feet,  or 
a  cord,  and  every  cord  contains  8  cord-feet;  and 
as  8  is  j'g  of  128,  every  cord-foot  contains  16  cubic 
feet ;  therefore,  dividing  the  cubic  feet  in  a  pile  of 
wood  by  16,  the  quotient  is  the  cord-feet;  and  if 
cord-feet  be  divided  by  8,  the  quotient  is  cords. 

Note. — If  we  wish  to  find  the  circumference  of 
a  tree,  which  will  hew  any  given  number  of  inches 
square,  we  divide  the  given  side  of  the  square  by 
.225,  and  the  quotient  is  the  circumference  re- 
quired. 

What  must  be  the  circumference  of  a  tree  that 
will  make  a  beam  10  inches  square  ? 

Note. — When  wood  is  "  corded"  in  a  pile  4  feet 
wide,  by  multiplying  its  length  by  its  hight,  and 
dividing  the  product  by  4,  the  quotient  is  the  cord- 
feet  ;  and  if  a  load  of  wood  be  8  feet  long,  and  its 
hight  be  multiplied  by  its  width,  and  the  product 
divided  by  2,  the  quotient  is  the  cord-feet. 

IIow  many  cords  of  wood  in  a  pile  4  feet  wide, 
70  feet  6  inches  long,  and  5  feet  3  inches  high? 

Note. — Small  fractions  rejected. 

To  find  how  large  a  cube  may  be  cut  from  any 
given  sphere,  or  be  inscribed  in  it. 

Rule. —  Square  the  diameter  of  the  sphere^  divicU 
that  product  hy  3,  and  extract  the  square  root  of  tht 
quotient  for  the  avswer. 


188      ORTON  &  Sadler's  caloulator. 

I  have  a  piece  of  timber,  30  inches  in  diameter, 
how  large  a  square  stick  can  be  hewn  from  it? 

Rule. — Multiply  the  diameter  by  .7071,  and  thi 
product  is  the  side  of  a  square  inscribed, 

I  have  a  circular  field,  360  rods  in  circumference; 
what  must  be  the  side  of  a  square  field  that  shall 
contain  the  same  quantity? 

Rule. — Multiply  the  circumference  by  .282,  and 
the  product  is  the  side  of  an  equal  square. 

I  have  a  round  field,  50  rods  in  diameter;  what 
is  the  side  of  a  square  field  that  shall  contain  the 
same  area?  Ans,  44.31 135-|- rods. 

Rule. — Midtiply  the  diameter  by  .886,  and  the 
product  is  the  side  of  an  equal  square. 

There  is  a  certain  piece  of  round  timber,  30 
inches  in  diameter ;  required  the  side  of  an  equi- 
lateral triangular  beam  that  may  be  hewn  from  it. 

Rule. — Multiply  the  diameter  by  .866,  a:xd  the 
product  is  the  side  of  an  inscribed  equilateral  tri- 
angle. 

To  find  the  area  of  a  globe  or  sphere. 

Definition. — A  sphere  or  globe  is  a  round  solid 
body,  in  the  middle  or  center  of  which  is  an  imag- 
inary point,  from  which  every  part  of  the  surface 
is  equally  distant.  An  apple,  or  a  ball  used  by 
ohildren  in  some  of  their  pastimes^  may  be  called 
a  sphere  or  globe. 


ROUND  TIMBER. 


Round  timber,  when  squared,  is  estimated  to 
lose  one-fifth ;  hence  (50  cubic  feet,  or)  a  ton  of 
round  timber  is  said  to  contain  only  40  cubic  feet. 

Round,  sawed,  and  hewn  timber  is  bought  and 
sold  by  the  cubic  foot. 

To  measure  round  timber. 

Rule.* — Take  the  girth  in  feet,  at  both  the  large 
and  small  ends,  add  them,  and  divide  their  sum  by 
two  for  the  mean  girth ;  then  multiply  the  length 
in  feet  by  the  square  of  one-fourth  of  the  mean 
girth,  and  the  quotient  will  be  the  contents  in  cubic 
''eet,  according  to  the  common  practice. 

*  This  rule  gives  ahont  four-ffths  of  the  true  contents,  oti0- 
yth  being  allowed  to  the  buyer  for  waste  in  hewing. 

189 


190      ORTON  <&  Sadler's  calculator. 

Example. — What  are  the  cubic  contents  of  a 
round  log  20  feet  long,  9  feet  girth  at  the  large 
end,  and  7  feet  at  the  small  end  ? 

Solution. — 9  +  7  =  16-5-2  =  8  mean  girth. 

Then  20  length  x  4  feet  (the  square  of  ^  mean 
girth)  ==  80  cubic  feet.     Ans. 

Note. — If  the  girth  be  taken  in  inches,  and  the 
length  in  feet,  divide  the  last  product  by  144. 

Example. — What  are  the  cubic  contents  of  a 
round  log  12  feet  long,  50  inches  girth  at  the  large 
end,  38  inches  at  the  small  end? 

Work.— 50  -f  38  =  88  -^  2  =  44  mean  girth. 

Then  12  length  x  121  inches  (the  square  of  ^ 
mean  girth)  =  1452  -:-  144  =  10  j^^  cubic  feet. 

To  measure  round  timber  as  the  frustum  of  a 
cone :  that  is,  to  measure  all  the  timber  in  the  log. 

Rule. — Multiply  the  square  of  the  circumference 
at  the  middle  of  the  log  in  feet  by  8  times  the  length, 
and  the  product  divided  by  100  will  be  the  contents. 
Extremely  near  the  truth. 

Note. — The  above  rule  makes  1  foot  more  timber 
in  every  190  cubic  feet  a  log  contains  if  ciphered 
out  by  the  long  and  tedious  rules  of  Geometry.  It 
Is  therefore  suflBciently  correct  for  all  practical  pur- 
poses, and  this  rule  being  so  short  and  simple  in 
comparison  with  all  others,  every  lumberman,  ship- 
builder, carpenter,  inspector  or  surveyor  of  timber, 
should  post  it  up  for  reference  and  use. 


TIMBER   MEASURE. 


191 


A  TABLE  FOR  MEASURING  TIMBER. 


Qnart«»r 
Oirt. 

Area. 

Quarter 
Girt. 

Area. 

Quarter 
Girt. 

Area. 

Inches. 

6 

6i 
6J 

Feet. 

.250 
.272 
.294 
.317 

Inches. 

12 

m 

12| 

Feet. 
1.000 

1.042 
1.085 
1.129 

Inches. 

18 
18} 
19 
19} 

Feet. 

2.250 
2.376 
2.506 
2.640 

7 

7i 
71 

.340 
.364 
.390 
.417 

13 
13i 
13} 
13i 

1.174 
1.219 
1.265 
1.313 

20 
20} 
21 
21} 

2.777 
2.917 
3.062 
3.209 

8 

8} 
8} 
8i 

.444 
.472 
.501 
.531 

14 
14^ 
14} 
14i 

1.361 
1.410 
1.460 
1.511 

22 

22} 

23 

23} 

3.362 
3.516 
3.673 
3.835 

9 
9i 

n 

.562 
.594 
.626 
.659 

15 

15J 
15} 
15| 

1.562 
1.615 
1.668 
1.722 

24 
24} 
25 
25} 

4.000 
4.168 
4.340      I 
4.516 

10 

10} 
10} 
lOJ 

.694 
.730 
.766 
.803 

16 

m 

16} 
16i 

1.777 
1.833 
1.890 
1.948 

26 
26} 
27 
27} 

4.694 
4.876 
5.062 
5.252 

11 
11} 

Hi 
iij 

.840 
.878 
.918 
.959 

17 

m 

17} 
171 

2.006 
2.066 
2.126 
2.187 

28 

28} 

29 

29} 

30 

5.444 
5.640 
5.840 
6.044 
6.250 

To  measure  round  timber  by  the  table. 
Multiply  the  area  corresponding  to  the  quarte^ 
girt  in  inches  by  the  length  of  the  log  in  feet 


192   ORTON  &  Sadler's  calculator. 


Note. — If  the  quarter-girt  exceed  the  table,  take 
half  of  it,  and  four  times  the  contents  thus  formed 
5jrill  be  the  answer. 

EXAMPLE  1. 

If  a  piece  of  round  timber  be  18  feet  long,  and 
the  quarter  girt  24  inches,  how  many  feet  of  timber 
are  contained  therein  ? 


24  quarter 
24 

girt. 

96 

48 

By  the  Table. 

576  square. 

18 

Against  24  stands    4.00 
Length,       18 

4608 

Product,                  72.00 

576 

An<»    72  fpptL 

144)10368(72  feet 
1008 

xXUS.     1  ii    ICCli* 

288 

288 

This  table  gives  the  oustomary,  but  only  about  foit>r -fifths  ol 
the  true  contents,  one-fifth  being  allowed  the  buyer  for  waste 
n  hewing  or  sawing  to  make  the  timber  square. 

The  following  rule  gives  the  true  contents  ; — 
Multiply  square  of  girth  by  .08  times  length. 
In  the  above  example  the  whole  gi/th  is  8  feet, 
squared  is  64  x  (.08  x  18  length)  ^  92.16  feet 


TIMBER   MEASURE.  193 

I.    Of  Flooring. 

Joists  are  measured  by  multiplying  their  breadth 
by  their  depth,  and  that  product  by  their  »ength 
They  receive  various  names,  according  to  the  posi- 
tion in  which  they  are  laid  to  form  a  floor,  such  as 
trimming  joists,  common  joists,  girders,  binding 
joists,  bridging  joists  and  ceiling  joists. 

Grirders  and  joists  of  floors,  designed  to  bear 
fi;reat  weights,  should  be  let  into  the  walls  at  each 
end  about  two-thirds  of  the  wall's  thickness. 

In  boarded  flooring,  the  dimensions  must  be 
taken  to  the  extreme  parts,  and  the  number  of 
squares  of  100  feet  must  be  calculated  from  thesd 
dimensions.  Deductions  must  be  made  for  stair- 
cases, chimneys,  etc. 

Example  1.  If  a  floor  be  57  feet  3  inches  long 
and  28  feet  6  inches  broad,  how  many  squarsa  oJ 
flooring  are  there  in  that  room  ? 


By  Decimals, 
57.25 

28.5 

28625 

By  Duodecimah 
F.         I. 
57    :    3 
28    :    6 

45800 
11450 

456 
114 
28    :    7    :    6 
7:0:0 

100)1631.625  feet. 

Squares  16.31625 

16:31  :    7    :    6 

Ana.  16  squares  and  31  feet 
17 


SQUARE  TIMBER 


To  measure  square  timber. 

IluLE. — Multiply  the  breadth  in  feet  by  the 
iepth  in  feet,  and  that  by  the  length  in  feet,  and 
the  quotient  will  be  the  contents  in  cubic  feet. 

Example. — How  many  cubic  feet  in  a  square  log 
12  feet  long  by  2  feet  broad  and  1^  feet  deep  ? 

Explanation. — 2  feet  breadth  x  1^  feet  depth 
X  12  feet  length  =  36  cubic  feet.     Ans. 

Note. — If  the  breadth  and  depth  be  taken  in 
inches,  divide  the  last  product  by  144. 

Example. — :How  many  cubic  feet  in  a  square  log 
24  feet  long,  30  inches  broad,  and  20  inches  deep  ? 

Solution. — 30  inches  breadth  x  20  inches  depth 
X  24  feet  length  =  14400  -h  144  -  100  cubic  feet 
194 


TIMBER  MEASURE.  195 

PROBLEM 
To  find  iht  solid  contents  of  squared  or  four-sided 
Timber, 
By  the  Carpenters  Rule, 
As  12  on  D  ;  length  on  c  :  Quarter  girt  on  D  : 
dolidity  on  0. 

Rule  I. — Multiply  the  breadth  in  the  middle  by 
the  depth  in  the  middle^  and  that  product  by  the 
length  for  the  solidity. 

Note. — If  the  tree  taper  regularly  from  one  end 
to  the  other,  half  the  sum  of  the  breadths  of  the 
two  ends  will  be  the  breadth  in  the  middle,  and 
half  the  sum  of  the  depths  of  the  two  ends  will  bo 
the  depth  in  the  middle. 

Rule  II. — Multiply  the  sum  of  the  breadths  of 
the  two  ends  by  the  sum  of  the  depths^  to  which  add 
the  product  of  the  breadth  and  depth  of  each  end ; 
one-sixth  of  this  sum  multiplied  by  the  length,  will 
give  the  correct  solidity  of  any  piece  of  squared  tim- 
ber tapering  regularly, 

PROBLEM 
To  find  how  much  in  length  will  make  a  solid  foot, 

or  any  other  asssigned  quantity,  of  squared  timber, 

of  equal  dimensions  from  end  to  end. 

Rule. — Divide  1728,  the  solid  inches  in  a  foot^ 
or  the  solidity  to  be  cut  off^  by  the  area  of  the  end 
in  inches,  and  the  quotient  will  be  the  length  in  inches 


196 


ORTON   &   SADLER  S   CALCULATOR. 


Note. — To  answer  the  purpose  of  the  above 
rule,  some  carpenters'  rules  have  a  little  table  upoD 
them,  in  the  following  form,  called  a  table  of  tim 
her  measure. 


0 

0 

0   0  1  9 

0  1  11  1  3  9  1  inches. 

144 
"T" 

36 

16  9  1  6 

4  1  2  1  2  1  1  feet. 

2 

3   4  1  5 

6  1  7  1  8  9  1  side  of  the  square. 

This  table  shows^  that  if  the  side  of  the  square 
be  1  inch,  the  length  must  be  144  feet ;  if  2  inches 
be  the  side  of  the  square,  the  length  must  be  3^6 
feet,  to  make  a  solid  foot. 

CAPACITY   OF   CISTERNS   OR   WELLS. 

Tabular  view  of  the  number  of  gallons  contained 
in  the  clear  between  the  brickwork  for  each  ten 
inches  of  depth : 


Diameter.  Gallons. 

2  feet  equal 19 

2i  "  "  

3  "  "  

3i  "  "  

4  "  " 

4i  "  ''  '^''ZZ 

5  "  " 
5i  "  "  'Z'''Z 

6  "  "  

Qh  "  "  

n  t(  It 

7i  "  "  'ZZ.Z 


30 
44 

8i 
9 

60 

9^ 

78 

10 

97 

11 

122 

12 

148 

13 

176 

14 

207 

15 

240 

20 

275 

25 

Diameter.  Gallons. 

8    feet  equal 313 

""     "        "    353 


461 

489 

692 

705 

827 

959 

1101 

1958 

3059 


CISTERNS   AND    RESEBVOIBS. 

How  to  measure  their  contents. 


1st.  Find  the  solid  contents  of  the  cistern  in  cubic  inches. 

2d.  Divide  the  contents  so  found  by  14553,  and  the  quo- 
tient will  be  the  number  of  hogsheads. 

If  the  height  of  the  cistern  be  given,  how  do  you  find 
the  diameter,  so  that  the  cistern  shall  contain  a  given 
number  of  hogsheads? 

1st.  Reduce  the  height  of  the  cistern  to  inches,  and  the 
contents  to  cubic  inches. 

2d.  Multiply  the  height  by  the  decimal  .7854. 

3d.  Divide  the  contents  by  the  last  result,  and  extract 
the  square  root  of  the  quotient,  which  will  be  the  diameter 
of  the  cistern  in  inches. 

Note.— In  tHtiniating  the  capacity  of  cisterns,  reservoir?,  etc.,  tb« 
fblluwing  table  is  ustd: 

31i  gallons 1  barrel. 

63        '*       1  hogshead. 

The  barrels  used  in  commerce  vary  from  30  to  45  gallon^ 
and  the  hogshead  from  40  to  tJO  gallons. 

197 


198    ORTON  &   SADLER'S  CALCULATOR. 

CISTERNS  AND  RESERVOIRS. 

HOW  TO  MEASURE  THEIR  CONTENTS. 

Cisterns  and  Reservoirs  are  constructed  for  the  purpose 
of  holding  large  quantities  of  water  or  fluids,  and  are  in  the 
form  of  a  tub  cylinder,  or  solid  square.  They  are  generally 
built  in  the  ground,  and  mortised  on  all  sides,  except  the 
opening,  with  brick  or  stone,  and  are  chiefly  permanent 
in  their  construction. 

TO   MEASURE  ROUND  OR  CYLINDRICAL  CISTERNS. 

Rule  I. — Multiply  the  square  of  the  diameter  in  feet  by 
the  depth  in  feet,  which  wilt  give  the  number  of  cylindrical 
feet  in  the  cistern. 

Rule  II. — Multiply  the  cylindrical  feet  by 

373 

for  Hogsheads. 

4000 

373 

—  for  Barrels,  or  divide  by  5. 
2000 

47 

—  for  Gallons. 
8 

The  result  will  be  the  contents  either  in  hogsheads,  barrels, 
or  gallons,  depending  upon  the  fractions  used. 

Example  I. — What  is  the  contents  in  hogsheads  of  a 
cistern  20  feet  in  diameter  and  10  feet  deep  ? 

Solution. — 20  ft.  X  20  ft.  =  400  ft.  square  of  diameter. 
400  "  X  10  "  =  4000  cylindrical  ft. 

373 

4000  cylindrical  ft.  X =z  373  hhds.— J«j». 

4000 

Example  II. — How  many  gallons  in  a  cistern  10  feet  in 
diameter  and  16  feet  deep? 


TO   MEASURE   CISTERNS.  199 

Solution.— 10  X  10  =  100  X  16  =  1600  cylindrical  ft 

47 

1600  X  —  =  9400  gallons.— ^W5. 


Or,  373 

1600  X  —  =  298f  barrels.— ^715. 
2000 

Or  373 

1600  X =  149  i  hogsheads.— ^n«. 

4000 

TO  MEASURE  SQUARE  CISTERNS. 

Rule  I. — Multiply  the  width  in  feet  hy  the  length  in  feet, 
and, that  2)roduct  by  the  depth  in  feet;  this  last  product 
will  be  the  number  of  cubic  feet  in  the  cistern. 

Rule  II. — Multiply  the  cubic  feet  thus  obtained  by 
19 

—  for  Hogsheads. 
160 

19 

—  for  Barrels. 

80 

7  -jYjj  for  Gallons. 

This  will  give  the  contents  either  in  hogsheads^  barrels,  or 
geUlonSj  as  desired. 

Example. — How  much  water  will  a  cistern  contain 
which  is  6  feet  wide,  8  feet  long,  and  10  feet  deep  ? 

Solution.— 6  X  8  =  48  X  10  =  480  cubic  ft.  in  cistern. 

19 
480  X  —  =  57  hogsheads. — Ans. 
160 

19 

Or,  480  X  —  =  114  barrels.— ^rw. 

80 

40 
Or,  480X  7  A8ff  or 480X7.48  =rr 3590-  -gaii.—^««. 

100 


OASK-GAUGING. 


ixauging  is  the  art  of  measuring  the  capacity  o{ 
casks  and  vessels  of  any  form.  In  commerce,  most 
of  the  gauging  is  done  by  the  use  of  the  diagonal 
rod,  which  gives  only  approximate  results,  but  suf- 
ficiently accurate  for  ordinary  purposes. 

Ullage  is  the  difference  between  the  actual  con- 
tents of  a  vessel  and  its  capacity,  or  that  part 
which  is  empty. 

To  measure  small  cylindrical  vessels. 

Rule. — Multiply  the  square  of  the  diameter,  in 
inches,  by  34,  and  that  by  the  height,  in  inches, 
and  point  off  four  figures ;  the  result  will  be  th< 
capacity,  in  wine  gallons  and  decimals  of  a  gallon 

For  beer  gallons  multiply  by  28  instead  of  34. 
200 


CASK-GAUGING.  201 

Example. — A  can  measures  15  inches  in  diame- 
ter, and  is  2  feet  2  inches  in  heiglit.  How  many  gal 
Ions  will  it  contain  ?  15x15  =  225  X  26  height  = 
5850 ;  5850  X  34  =  19.8900.     Ans.  19yVu  galls. 

Casks  are  usually  regarded  as  the  two  equal  frus- 
tums of  a  cone,  and  are  very  accurately  gauged  by 
three  dimensions  as  follows  : — 

To  measure  a  cask  by  three  dimensions. 

1st.  Add  the  bung  and  head  diameters  in  inches, 
and  divide  by  2  for  the  mean  diameter. 

2d.  Multiply  the  square  of  the  mean  diameter  by 
the  length  of  the  cask  in  inches. 

3d.  Multiply  the  last  product  by  .0034  for  wine 
gallons,  .0028  for  beer  gallons. 

Example. — How  many  wine  gallons  in  a  cask, 
the  bung  diameter  of  which  is  22  inches,  the  head 
diameter  20  inches,  and  the  length  32  inches  ? 

Work.— 22  -f  20  =  42  -t-  2  =  21  (mean  diame- 
ter) :  then  21  x  21  =  441  (square  of  mean  diame 
ter),  x  32  length  =  14112  x  .0034  =  47.9808. 
Ans. 

Note. — If  the  cask  is  not  full,  stand  it  on  the 
end,  and  multiply  by  the  height  of  the  liquid,  in- 
stead of  the  length  of  the  cask,  for  actual  contents 

When  the  cask  is  much  bilged  or  rounded  front 
the  bung  to  the  head,  a  more  accurate  way  is  to 
gauge  by  four  iimcnsions,  as  follows : — 


202      ORTON  &  Sadler's  calculator. 

To  measure  a  cask  by  four  dimensions, 

1st.  Add  the  bung  and  head  diameters  in  inches, 
and  the  diameter  in  inches  between  bung  and  head. 

2d.  Divide  their  sum  by  3  for  the  mean  diameter. 

3d.  Multiply  the  square  of  the  mean  diameter 
by  the  length  of  the  cask  in  inches. 

4*th.  Multiply  the  last  product  by  .0034  for  wine 
gallons,  .0028  for  beer  gallons. 

Example. — What  are  the  contents  in  gallons  of 
a  cask,  the  bung  diameter  of  which  is  24  inches, 
the  middle  diameter  20  inches,  the  head  diameter 
16  inches,  and  its  length  40  inches? 

Work.— 24  +  20  -f  16  =  60  -r-  3  =  20  (mean 
diameter),  then  20  X  20  =  400  (square  of  mean  dia- 
meter) X  40  length  =  16000  x  .0034  =  54. 4  gallons 

1.  The  ale  gallon  contains  282  cubic  inches. 

2.  The  wine  gallon  contains  231  cubic  inches. 
3    The  bushel  contains  2150.4  cubic  inches. 

4.  A  cubic  foot  of  pure  water  weighs  1000 
ounces  =  62^  pounds  avoirdupois. 

5.  To  find  what  weight  of  water  may  be  put  into 
a  given  vessel. 

Multiply  the  cubic  feet  by  1000  for  the  ounces, 
or  by  6 2^ /or  the  pounds  avoirdupois. 

6.  What  weight  of  water  can  be  put  irto  a  cis- 
icrn  ti  feet  square  ?     Ans,  26,367  lbs.  3  o% 


MEASURING  GRAIN. 


By  the  United  States  standard,  2150  cubic  inchef 
make  a  bushel.  Now,  as  a  cubic  foot  contain? 
TT28  cubic  inches,  a  bushel  is  to  a  cubic  foot  as 
2150  to  1728;  or,  for  practical  purposes,  as  4  to 
5.  Therefore,  to  convert  cubic  feet  to  bushels,  it 
is  necessary  only  to  multiply  by  |.or  8. 

To  measure  the  bushels  of  grain  in  a  granary 

Rule. — Multiply  the  length  in  feet  by  tht 
breadth  in  feet,  and  that  again  by  the  depth  in 
feet,  and  that  again  by  J.  The  last  product  will 
be  the  number  of  bushels  tJie  granary  contains. 

Example. — How  many  bushels  in  a  bin  10  feet 
long,  4  feet  wide,  and  4  feet  deep. 

Work. — 10  feet  length  X  4  feet  breadth  x  4  fee/ 
depth  a.  IGO  <'ubic  fe^t ;  then  IBO  x  J  =  128.  Ans. 
203 


204      ORTON  &  Sadler's  calculator. 

SIZE  OF  BINS 

To  contain  a  given  number  of  bushels. 

Having  any  number  of  bushels,  how  then  will 
jcu  find  the  corresponding  number  of  cubic  feet? 

Increase  the  number  of  bushels  one-fourth  itself, 
and  the  result  will  be  the  number  of  cubic  feet. 

How  will  you  find  the  number  of  bushels  which 
a  bin  of  a  given  size  will  hold  ? 

Find  the  content  of  the  bin  in  cubic  feet ;  then 
diminish  the  content  by  one-fifth,  and  the  result 
will  be  the  content  in  bushels. 

How  will  you  find  the  dimensions  of  a  bin 
which  shall  contain  a  given  number  of  bushels  ? 

Increase  the  number  of  bushels  one-fourth 
itself,  and  the  result  will  show  the  number  of 
cubic  feet  which  the  bin  will  contain.  Then, 
when  two  dimensions  of  the  bin  are  known, 
divide  the  last  result  by  their  product,  and  the 
quotient  will  be  the  other  dimension. 

If  you  wish  the  contents  of  a  pile  of  ears  of 
corn,  or  roots,  in  heaped  bushels,  ascertain  the 
cubic  feet,  and  multiply  by  iVu. 


WEIGHTS  AND  MEASURES 

Recognized  by  the  Laws  of  the  United  States, 

In  some  States  dried  apples  and  peaches  are 
sold  by  the  heaping  bushel  as  are  other  of  farm 
products. 

A  bushel  of  corn  in  the  ear  is  three  heaped 
half-bushels,  or  four  even-full. 


TABLE  OF  AVOIRDUPOIS  POUNDS  ITI  A  BUSHEL, 

As  prescribed  by  statute  in  the  several  States  named. 


'mi 

IOC    •    -(N    -oooooco    'CO    •    -(M    -co    • 
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:8  : 

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t-aM^  :  :§  .^  :  :S  :§  :  :^  :SS  : 

::S: 

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•«".wl^  :  :2i  :gS25 

:ig  :  \^  '  :S  : 

:8  : 

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p 

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1                                                                 X' 

'^^bs::^  :3  :  :%t'^  :g§J;^8f§  :8i5S?, 

•««*<'/ |2^83S?i?8^S5S5f§S  :^{HSr§  :^i§85 

•/>/(/- 

2S8-.^ 
^82  = 

^8S??ig^§g8?^:^8S  :§58  :  | 

7/7 

^§jSfS§:^S^^?^:^8:;;  : 

12  -  ^ 

•«uoj  1   :  :  -.-t?  :  :  :  :  :  :s  :  :g5  :So  : 

•  o     • 

7»J|§  ::§:::::  :S  ::??:  ::^  : 

:8  : 

H 

Q 

8 

:  :  : 

Barley 

Beans 

Bine  Grass  Seed... 
Bnckwlieat 

Castor  Beans 

Clover  Seed 

Dried  Apples 

Dried  Peaches 

Flax  Seed 

IlenipSeed 

Indian  Corn 

Indian  (  om  in  ear 
Indian  Corn  Meal.. 
Oafs 

il 

5^^ 

1 

1  :i 

111 
1^^ 

In  Pennsylvania  80  lbs  coarse,  70  lbs.  ground,  or  62  lbs.  fine  salt 
make  1  bnshel ;  and  in  Illinois,  50  lbs.  common,  or  55  lbs.  fine  salt 
make  1  bushel.  In  Tennessee  100  ears  of  coiu  are  a  bushel.  A 
heaping  bushel  contains  2S15  cubic  inches. 

In  Maine  64  lbs.  of  ruta  baga  turnips  or  beets  make  1  bushel. 

A  cask  of  lime  is  240  lbs  Lime  in  slacking  absorbs  2)^  times  its 
volume,  and  2>^  times  its  weight  in  water. 


18 


205 


RAILROAD    FREIGHT. 

TABLE  OF  GROSS  WEIGHTS. 

The  Articles  named  are  Billed  at  actual  weights,  if  possible, 
but  usually  at  the  weights  in  the  Table  below  when  it  is 
not  convenient  to  weigh  them..  • 

Ale  and  Beer 320  lb.  per  bbl. 


"    170 

**  "    100 

Apples,  dried 24 

"        green 56 

«  "     150 

Barley 48 

Beans,  wliilo 60 

"      castor 46 

Beef 320 

Bran 20 

Brooms 40 

Buckwheat 52 

Cider 350 

Charcoal 22 

Clover  Seed 60 

Corn 56 

"   in  car 70 

"    Meal 48 

"       «    220 

Eggfi 200 

FvA\ 300 

Flax  Seed 56 

Flour 200 

Hemp  Seed 44 


K' 


bbl. 
bu. 


bbl. 
bu. 
doz. 
bu. 
bbl. 
bu. 


bbl. 


bu. 
bbl. 
bu. 


High  wines 350  lb.  per  bbl. 

Hungarian    Grass 

Seed 45  "  bu. 

Lime 200  "  bbl. 

Malt 38  "  bu. 

Millet 45  " 

Nails 108  "  kog. 

Oats 32  "  bu. 

Oil 400  "  bbl. 

Onions 57  '*  bu. 

Peaches,  dried. . .       33  "  *' 

Pork 320  "  bbl. 

Potatoes,  common.  150  "  " 

.    60  "  bu. 

"        sweet. . .     55  "  " 

Rye 56  «  " 

Salt,  fine 56  *  « 

"       " 300 


"    coarse 

"    in  sacks.. 

Timothy  Seed. 

Turnips 

Vinegar 

Wheat 

Whiskey 


350 

203 
4") 
56 

350 
60 

350 


bbl. 

sack, 
bu. 

bbl. 
bu. 
bbl. 


One  ton  weight  is  2000  lbs. 

ESTIMATED  WEIGHTS  OF  lUMBER  AND  OTHER  ARTICLES. 

Note.— From  18,000  to  20,000  lb.  is  considered  a  car-load  in  most 
places,  each  car  itself  also  weighing  about  20,000  lb. 

206 


rO  MEASURE  CORN  ON  THE  COB  IN  CRIBS 


Corn  is  generally  put  up  in  cribs  made  of  rails ; 
but  the  rule  will  apply  to  a  crib  of  any  size  or  kind, 
whether  equilateral,  or  flared  at  the  sides. 

When  the  crib  is  equilateral. 

Rule.  —  Multiply  the  length  in  feet  by  the 
breadth  in  feet,  and  that  again  by  the  height  in 
feet,  which  last  product  multiply  by  .63  (the  frac- 
tional part  of  a  heaped  bushel  in  a  cubic  foot),  and 
the  result  will  be  the  heaped  bushels  of  ears.  For 
the  number  of  bushels  of  shelled  corn  multiply  by 
42  (two-thirds  of  .63),  instead  of  .63, 
207 


208      ORTON  &  Sadler's  calculator. 

Example. — Required  the  number  of  bushels  of 
shelled  corn  contained  in  a  crib  of  ears,  15  feet 
long,  by  5  feet  wide,  and  10  feet  high  ? 

15  length  x  5  width,  x  10  height  =  150  cubic 
feet.  Then  Y50  X  .63  =  412.50  heaped  bushels  A 
ears.  Also  750  X  .42  =  315  bushels  of  shelled 
corn. 

In  measuring  the  height,  of  course,  the  height  of 
the  corn  is  intenaed.  And  there  will  be  found  to 
be  a  difference  in  measuring  corn  in  this  mode,  be- 
tween fall  and  spring,  because  it  shrinks  very  much 
in  the  winter  and  spring,  and  settles  down. 

Wlien  the  crib  zs  flared  at  the  sides. 

Rule. — Multiply  half  the  sum  of  the  top  and  bot- 
tom widths  in  feet  by  the  perpendicul-ar  height  in 
feet,  and  that  again  by  the  length  in  feet,  which 
last  product  multiply  by  .63  for  heaped  bushels  of 
ears,  and  by  .42  for  the  number  of  bushels  of 
shelled  corn. 

Note. — The  above  rule  assumes  that  three  heap- 
ing hclf  bushels  of  ears  make  one  struck  bushel 
of  shelled  corn.  This  proportion  has  been  adopted 
upon  the  authority  of  the  major  part  of  our  best 
agricultural  journals.  Nevertheless,  some  journals 
claim  that  two  heaping  bushels  of  ears  to  one  of 
shelled  corn  is  a  more  correct  proportion,  and  it  is 
the  custom  in  many  parts  of  the  country  to  buy 


MEASUREMENT   OF   CORN.  209 

and  sell  at  that  rate.  Of  course  much  will  de- 
pend upon  the  kind  of  corn,  the  shape  of  the  ear, 
the  size  of  the  cob,  &c.  Some  samples  are  to  be 
found,  three  heaping  half  bushels  of  which  will 
even  overrun  one  bushel  shelled ;  while  others 
again  are  to  be  found,  two  bushels  of  which  will  fall 
short  of  one  bushel  shelled.  Every  farmer  must 
judge  for  himself,  from  the  sample  on  hand,  whether 
to  allow  one  and  a  half  or  two  bushels  of  ears  to 
one  of  shelled  corn.  In  either  case,  it  is  onl^  an 
approximate  measurement,  but  sufficient  for  all  ordi- 
nary purposes  of  estimation.  The  only  true?  vay 
of  measuring  all  such  products  is  by  weight 


MEASUREMENT  OF  HAY- 


The  only  correct  mode  of  measuring  hay  is  to 
«veigh  it.  This,  on  account  of  its  bulk  and  cha- 
racter, is  very  difficult,  unless  it  is  baled  or  other- 
wise compajted.  This  difficulty  has  led  farmers 
to  estimate  the  weight  by  the  bulk  or  cubic  con- 
tents, a  mode  which  is  only  approximately  correct. 
Some  kinds  of  hay  are  light,  while  others  are 
heavy,  their  equal  bulks  varying  in  weight.  Bui 
for  all  ordinary  farming  purposes  of  estimating  the 
amount  of  hay  in  meadows,  mows,  and  stacks,  the 
following  rules  will  be  found  sufficient : — 

As  nearly  as  can  be  ascertained,  25  cubic  yards 
of  average  meadow  hay,  in  windrows,  make  a  ton 
210 


MEASUREMENT  OF  HAY. 


211 


When  loaded  on  wag(  ns,  or  stored  in  barns,  20 
cubic  yards  make  a  ton. 

When  well  settled  in  mows,  or  stacks,  15  cubic 
yards  make  a  ton. 

Note. — These  estimates  are  for  mecfium-eii  t%i 
mows  or  stacks ;  if  the  hay  is  piled  to  a  great 
height,  as  it  often  is  where  horse  hay-forks  are 
used,  the  row  will  be  much  heavier  per  cubic  yard. 

When  hay  is  haled,  or  closely  packed  for  ship- 
ping, 10  cubic  yards  will  weigh  a  ton. 

To  find  the  number  of  tons  in  long  square  stacks. 

Rule. — Multiply  the  length  in  yards  by  the 
width  in  yards,  and  that  by  half  the  altitude  in 
yards,  and  divide  the  product  by  15. 

Example. — How  many  tons  of  hay  in  a  square 
stack  10  yards  long,  5  wide,  and  9  high  ? 

Solution.— 10  X  5  X  4^  =  225-^  15  =  15  tons. 
Ans. 

To  find  the  number  of  tons  in  circular  stacks. 

Rule. — Multiply  the  square  of  the  circumference 
in  yards  by  4  times  the  altitude  in  yards,  and  di- 
vide by  100 ;  the  quotient  v/ill  be  the  number  of 
cubic  yards  in  the  stack ;  then  divide  by  15  for  the 
number  of  tons. 

Example. — How  many  tons  of  hay  in  a  circular 
Btack,  whose  circumference  at  the  base  is  25  yards, 
and  height  9  vards  ? 


212      ORTON  A  Sadler's  calculator. 

Solution. — 25  x  25  =  625,  the  square  of  the 
circumference ;  then  625  x  36  (four  times  the 
length),  =  225000  ^  100  =  225  (the  number  oi 
cubic  yards),  then  225 -r- 15  =  15,  the  number  of 
tons. 

An  easy  mode  of  asceiHaining  the  value  of  a 
given  number  of  lbs,  of  hay,  at  a  given  price  per 
ton  of  2000  lbs. 

Rule. — Multiply  the  number  of  pounds  of  hay 
(coal,  or  anything  else  which  is  bought  and  sold 
by  the  ton)  by  one-half  the  price  per  ton,  pointing 
off  three  figures  from  the  right  hand  ;  the  remain- 
ing figures  will  be  the  price  of  the  hay  (or  any 
article  by  the  ton). 

Example. — What  will  658  lbs.  of  hay  cost,  @ 
$T  50  per  ton  ? 

Solution. — $7  50  divided  by  2  equals  $3  75,  by 
which  multiply  the  number  of  ponnlg,  thus  :  658  x 
$3  75  =  246.750,  or  $2  46.     An, 

Note. — The  principle  in  this  rule  is  the  same  as 
in  interest — dividing  the  price  by  two  gives  us  the 
price  of  half  a  ton,  or  1000  lbs.  ;  and  pointing  jfl 
three  figures  to  the  right  is  dividing  by  1000. 

A  truss  of  hay,  new,  is  60  lbs.  ;  old,  50  lbs. ; 
straw,  40  lbs. 

A  load  of  hay  is  36  trusses. 

A  bale  of  hay  is  300  lbs. 


RULES  FOR  DETERMINING  THE  WEIGHT 
OF  LIVE  CATTLE. 


For  cattle  of  a  girth  of  from  5  to  Y  feet,  allow 
23  lbs.  to  the  superficial  foot. 

For  cattle  of  a  girth  of  from  t  to  9  feet,  allow 
31  lbs.  to  the  superficial  foot. 

For  small  cattle  and  calves  of  a  girth  of  from  3 
to  5  feet,  allow  16  lbs.  to  the  superficial  foot. 

For  pigs,  sheep,  and  all  cattle  measuring  less  than 
3  feet  girth,  allow  11  lbs.  to  the  superficial  foot 

Measure  in  inches  the  girth  round  the  breast, 
just  behind  the  shoulder-blade,  and  the  length  of 
the  back  from  the  tail  to  the  forepart  of  the  shoui- 
der-blade  Multiply  the  girth  by  the  length,  and 
divide  by  144  for  the  superficial  feet,  and  then  n  ul- 
213 


214      ORTON  &  Sadler's  calculator. 

tiplj  the  superficial  feei  by  the  number  of  lbs, 
allowed  for  cattle  of  different  girths,  and  the  pro- 
duct will  be  the  number  of  lbs.  of  beef,  veal,  or  pork, 
in  the  four  quarters  of  the  animal.  To  find  the 
number  of  stone,  divide  the  number  of  lbs.  by  14. 

Example — What  is  the  estimated  weight  oi 
beef  in  a  steer,  whose  girth  is  6  feet  4  inches,  and 
length  5  feet  3  inches  ? 

Solution. — 76  inches  girth,  x  63  inches  length, 
=  4788  -r-  144  =  33J  square  feet,  X  23  =  764| 
lbs.,  or  54|  stone.     Ans, 

Note. — When  the  animal  is  but  half  fattened,  a 
deduction  of  one  lb.  in  every  20  must  be  made; 
and  if  very  fat,  one  lb.  for  every  20  must  be  added. 

Where  great  numbers  of  cattle  are  annually 
bought  and  sold  under  circumstances  that  forbid 
ascertaining  their  weight  with  positive  accuracy, 
the  estimated  weight  may  be  thus  taken  with  ap- 
proximate exactness — at  least  with  as  much  accu- 
racy as  is  necessary  in  tlie  aggregate  valuation  of 
stock.  No  rules  or  tables  can,  however,  be  at  all 
times  implicitly  relied  on,  as  there  are  many  cir* 
cumstances  connected  with  the  build  of  the  animal, 
the  mode  of  fattening,  its  condition,  breed,  &c.. 
that  will  influence  the  measurement,  and  conse* 
quently  the  weight.  A  person  skilled  in  estimate 
irig  the  weiixht  of  stock  soon  learns,  however,  to 
make  allowance  for  all  these  circumstances. 


BRICK  BUILDING. 


A  perch  of  stone  is  24.  Y5  cubic  feet ;  when  bnilt 
In  the  wall,  22  cubic  feet  make  1  perch,  2J  cubic 
feet  being  allowed  for  the  mortar  and  filling. 

Threo  pecks  of  lime  and  four  bushels  of  sand  to 
a  perch  of  wall. 

To  find  the  number  of  perches  of  stone  in  walls. 

Rule. — Multiply  the  length  in  feet  by  the  height 
in  feet,  and  that  by  the  thickness  in  feet,  and  divide 
tlie  product  by  22. 

Example. — How  many  perches  of  stone  con- 
tained in  a  wall  40  feet  long,  20  feet  high,  and  18 
inches  thick  ? 

Solution. — 40  feet  length  x  20  feet  height  x  ^ 
feet  thick  =  1200 .4-  22  «  54.54  perches.     Anf^. 
215 


216      ORTON  &  Sadler's  calculator. 

Note. — To  find  the  perches  of  masonry,  divide 
the  cubic  feet  by  24.75,  instead  of  22. 

Brick'ivorJc, 

The  dimensions  of  common  bricks  are  from  71 
to  8  inches  long,  by  41  wide,  and  2|  thick.  Front 
bricks  are  8i  inches  long,  by  4J  wide,  and  2}  tliick. 

The  usual  size  of  fire-bricks  is  9  J  inches  long,  by 
4f  wide,  by  2f  thick. 

22  to  23  common  bricks  to  a  cubic  foot  when 
laid ;  15  common  bricks  to  a  foot  of  8-inch  wall 
when  laid. 

To  find  the  nmnher  of  common  bricks  in  a  wall. 

Rule. — Multiply  the  length  of  the  wall  in  feet 
by  the  height  in  feet,  and  that  by  its  thickness  in 
feet,  and  that  again  by  22. 

Example. — How  many  common  bricks  in  a  w^all 
40  feet  long  by  20  feet  high,  and  12  inches  thick? 

Solution.— 40  feet  length  X  20  feet  height,  X  1 
foot  thick,  X  22  ==  17,600.     Ans. 

Note. — For  walls  8  inches  thick,  multiply  the 
length  in  feet  by  the  height  in  feet,  and  that  by  15. 

When  the  wall  is  perforated  by  doors  ard  win- 
iows,  deduct  the  sum  of  their  cubic  feet  from  the 
;ubic  contents  of  the  wall,  including  the  openings, 
aefore  multiplying  by  22  or  15  as  before. 

Laths. 

Laths  are  l\io\h  inches  wide,  by  4  feet  long,  are 
usually  set  i  inch  apart,  and  a  bundle  contains  100. 


bricklayers'  work.  217 

BBICKLAYEBS'  WORK. 

The  principal  is  tiling,  slating,  walling  and  chim- 
oey  work. 

Of  Tiling  or  Slating, 

Tiling  and  slating  are  measured  by  the  square 
3f  100  feet,  as  flooring,  partitioning  and  roofing 
were  in  the  Carpenters'  work ;  so  that  there  is  not 
much  difference  between  the  roofing  and  tiling ; 
yet  the  tiling  will  be  the  most ;  for  the  bricklayers 
sometimes  will  require  to  have  double  measure  for 
hips  and  valleys. 

When  gutters  are  allowed  double  measure,  the 
way  is  to  measure  the  length  along  the  ridge- tile, 
and  add  it  to  the  content  of  the  roof:  this  makes 
an  allowance  of  one  foot  in  breadth,  the  whole 
length  of  the  hips  or  valleys.  It  is  usual  also  to 
allow  double  measure  at  the  eaves,  so  much  as  the 
projector  is  over  the  plate,  which  is  commonly 
about  18  or  20  inches. 

Sky-lights  and  chimney  shafts  are  generally  de- 
ducted, if  they  be  large,  otherwise  not. 

Example  1.  There  is  a  roof  covered  with  ti.es, 
whose  depth  on  both  sides  (with  the  usual  alt  iw* 
ance  at  the  eaves)  is  37  feet  3  inches,  and  che 
length  45  feet ;  how  many  squares  of  tiling  ar^ 
oontained  therein  ? 
19 


218      ORTON  &  Sadler's  calculator. 


BY 

DUODECIMALS 

FKET. 

INCHES. 

37 

3 

45 

0 

185 

148 

11 

3 

BY   DECIMALS. 
37.25 

45 

18625 
14900 


16  76.25 


16  76    3 

2.   Of  Walling. 

Bricklayers  commonly  measure  their  work  by 
the  rod  of  16 J  feet,  or  272 J  square  feet.  In  some 
places  it  is  a  custom  to  allow  18  feet  to  the  rod  ; 
that  is,  324  square  feet.  Sometimes  the  work  is 
measured  by  the  rod  of  21  feet  long  and  3  feer 
high,  that  is,  63  square  feet ;  and  then  no  regard 
is  paid  to  the  thickness  of  the  wall  in  measuring* 
but  the  price  is  regulated  according  to  the  thick- 
ness. 

When  you  measure  a  piece  of  brick-work,  the 
first  thing  is  to  inquire  by  which  of  these  ways  it 
must  be  measured;  then,  having  multiplied  the 
length  and  breadth  in  feet  together,  divide  the  pro- 
duct by  the  proper  divisor,  viz.:  272.25,  324  or  63, 
according  to  the  measure  of  the  rod,  and  the  quo- 
tient will  be  the  answer  in  square  rods  of  that 
measure. 

But,  commonly,  brick  walls  that  are  measured 
by  the  rod  are  to  be  rodnced  to  a  standard  thick- 


bricklayers'  work.  219 

Dess  of  a  brick  and  a-half,  which  may  be  done  by 
the  following 

Rule. — Multiply  the  number  of  superficial  feet 
that  are  contained  in  the  wall  by  the  number  of 
half  bricks  which  that  wall  is  in  thickness;  one- 
third  part  of  that  product  will  be  the  content  vi 
feet 

The  dimensions  of  a  building  are  generally 
taken  by  measuring  half  round  the  outside  and 
half  round  the  inside,  for  the  whole  length  of  the 
wall ;  this  length,  being  multiplied  by  the  hight, 
gives  the  superficies.  And  to  reduce  it  to  the 
standard  thickness,  etc.,  proceed  as  above.  All  the 
vacuities,  such  as  doors,  windows,  window  backs, 
etc.,  must  be  deducted. 

To  measure  any  arched  way,  arched  window  or 
door,  etc.,  take  the  hight  of  the  window  or  dooi 
trom  the  crown  or  middle  of  the  arch  to  the  bot- 
tom or  sill,  and  likewise  from  the  bottom  or  sill  to 
the  spring  of  the  arch ;  that  is,  where  the  arch 
begins  to  turn.  Then  to  the  latter  hight  add  twice 
the  former,  and  multiply  the  sum  by  the  width  of 
the  window,  door,  etc.,  and  one-third  of  the  pro- 
duct will  be  the  area,  sufficiently  near  for  practice. 

Example  1.  If  a  wall  be  72  feet  6  inches  long, 
and  19  feet  3  inches  high,  and  5 J  bricks  thick, 
how  many  rods  of  brick  work  are  contained  therein, 
when  reduced  to  the  standard  ? 


220      ORTON  &  Sadler's  calculator. 


glaziers'  wobk. 

GlajBiera  take  their  dimensions  in  feet,  inches 
iod  eights  or  tenths,  or  else  in  feet  and  hundredth 
parts  of  a  foot,  and  estimate  their  work  by  the 
i*quare  foot. 

Windows  are  sometimes  measured  by  taking  the 
dimensions  of  one  pane,  and  multiplying  its  super- 
ficies by  the  number  of  panes.  But,  more  gen- 
erally, they  measure  the  length  and  breadth  of  the 
window  over  all  the  panes  and  their  frames  for  the 
length  and  breadth  of  the  glazing. 

Circular  or  oval  windows,  as  fan  lights,  etc.,  are  . 
measured  as  if  they  were  square,  taking  for  their 
dimensions   the  greatest  length  and  breadth,  as  a 
compensation  for  the  waste  of  glass  and  labor  in 
cutting  it  to  the  necessary  forms. 

Example  1.  If  a  pane  of  glass  be  4  feet  8| 
inches  long,  and  1  foot  4^  inches  broad,  how  many 
feet  of  glass  are  in  that  pane  ? 


'  DUODECIMALS 

. 

BY  DECIMALS. 

FT.    IN.     P. 

4.729 

4     8     9 

1.354 

1     4     3 

18916 
23645 

4    8     9 

1     6  11 

0 

14187 

1     2 

2 

3 

4729 

6    4  10 

2 

3 

6.403066 

PLUMBERS     WORK. 


221 


PLUMBERS    WORK. 

Plumbers*  work  is  generally  rated  at  so  much 
per  pound,  or  by  the  liundred  weight  of  112 
pounds,  and  the  price  is  regulated  according  to  the 
value  of  lead  at  the  time  when  the  work  is  per- 
formed. 

Sheet  lead,  used  in  roofing,  guttering,  etc., 
weighs  from  6  to  12  pounds  per  square  foot,  ac- 
cording to  the  thickness,  and  leaden  pipe  varies  in 
weight  per  yard,  according  to  the  diameter  of  its 
bore  in  inches. 

The  following  table  shows  the  weight  of  a  square 
foot  of  sheet  lead,  according  to  its  thickness,  reck- 
oned in  parts  of  an  inch,  and  the  common  weight 
of  a  yard  of  leaden  pipe  corresponding  to  the 
diameter  of  its  bore  in  inches: 


Thickness 
of  Lead. 

Pounds  to  a 
Square  foot. 

Bore  of 
Leaden   Pipe. 

Pounds 
per  yard. 

5't 

5.899 

i 

10 

6.554 

1 

12 

1 

7.373 

H 

16 

1 

8.427 

li 

18          1 

9.831 

If 

21 

1 

11.797 

2 

- 

222 


ORTON    &   SADLER  8    CALCULATOR. 


Example  1.  A  piece  of  sheet  lead  measures  16 
feet  9  inclief?  in  length,  and  6  feet  6  inches  in 
breadth ;  what  is  its  weight  at  8 J  pounds  to  a 
square  foot  ? 


BY  DUODECIMA 

FEET.  INCHES. 

16         9 
6         6 

LS 

6 

BY  DECIMALS 
FEET. 

16.75 
6.5 

100 
8 

6 

4 

8375 
10050 

108 

10 

6 

108.875  feet. 

Then   1   foot   :    8J   pounds  :  :  108.875   feet 
898.21875  pounds=:8  cwt.  2 J  pounds  nearly. 


MASON  8    WORJL 


Masons  measure  their  work  sometimes  by  the 
foot  solid,  sometimes  by  the  foot  superficial,  and 
sometimes  by  the  foot  in  length.  In  taking 
dimensions  they  girt  all  their  moldings  as 
joiners  do. 

The  solids  consist  of  blocks  of  marble,  stone 
pillars,  columns,  etc.  The  superficies  are  pave- 
ments, slabs,  chimney-pieces,  etc. 


masons'  work.  223 

PLASTERERS     WORK. 

Plasterers'  work  is  principally  of  two  kinds, 
namely,  plastering  upon  laths,  called  ceiliny^  and 
plastering  upon  walls  or  partitions  made  of  framed 
timber,  called  rendering. 

In  plastering  upon  walls,  no  deductions  are  mado 
except  for  doors  and  windows,  because  cornices, 
festoons,  enriched  moldings,  etc.,  are  put  on  aft^r 
the  room  is  plastered. 

In  plastering  tiriiber  partitions,  in  large  ware- 
houses, etc.,  where  several  of  the  braces  and  larger 
timbers  project  from  the  plastering,  a  fifth  part  is 
commonly  deducted.  Plastering  between  their 
timbers  is  generally  called  rendering  between 
•juarters. 

Whitening  and  coloring  are  measured  in  tho 
same  manner  as  plastering ;  and  in  timbered  par- 
titions, one-fourth,  or  one-fifth  of  the  whole  area  is 
commonly  added,  for  the  trouble  of  coloring  the 
sides  of  the  quarters  and  braces. 

Plasterers'  work  is  measured  by  the  yard  square, 
consisting  of  nine  square  feet.  In  arches,  the  girt 
round  them,  multiplied  by  the  length,  will  give  the 
superficies. 

Example  I. —  If  a  ceiling  be  59  feet  6  inches 
long,  and  24  feet  6  inches  broad ;  how  many  yards 
does  that  ceiling  contain  ? 


224      ORTON  <fe  Sadler's  calculator. 

PROBLEM  L 

To  fiiid  the  solid  content  of  a  Dome^  having  the 
hight  and  the  dimensions  of  its  hose  given, 

UuLE. — Multiply  the  area  of  the  base  by  the 
hight ^  and  f  of  the  product  will  be  the  solidity. 

Example  1. — What  is  the  solidity  of  a  dome,  in 
the  form  of  a  hemisphere,  the  diameter  of  the  cir- 
cular base  being  60  feet  ? 

60'X -7854=2827.44  area  of  the  base. 

Then  f  (2827.44X30)=56548.8  cubic  feet, 
Ans, 

PROBLEM  n. 

To  find  the  superficies  of  a  dome^  having  the  high, 
and  dimensions  of  its  base  given. 

Rule. — Multiply  the  area  of  the  base  by  2,  and 
ihe  product  will  be  the  superficial  content  required  ; 
or^  multiply  the  square  of  the  diameter  of  the  base 
hy  1.5708. 

For  an  Elliptical  Dome. — Multiply  the  two 
diameters  of  the  base  together ^  and  that  product  by 
1.5708,  the  last  product  will  be  the  area^  sufficiently 
correct  for  practical  purposes. 


8H0RT  RULES  FOR  THE  MECHANIC. 


Question. — A  stick  of  timber  is  carried  by  three 
men,  one  carries  at  the  end,  and  the  other  two  with 
a  lever,  How  far  should  the  lever  be  placed  from 
the  other  end,  that  each  man  may  carry  equally  ? 

Rule. — Divide  the  length  of  the  stick  by  4,  and 
the  quotient  is  the  answer. 

There  is  a  stick  of  timber,  30  feet  long,  to  be 
carried  by  3  men :  one  carries  at  the  end,  the  other 
two  carry  by  a  lever ;  how  far  must  the  lever  be 
placed  from  the  other  end,  that  each  may  carry 
Hjually  ?  Ans.  7^  feet  from  the  end, 
225 


ISDUARE  S:C1IBE  RnnTS 


To  work  the  square  and  cube  roots  with  ease  and 
Tacility,  the  pupil  must  be  familiar  with  the  follow- 
[ufr  properties  of  numbers: 

Their  importance  can  not  be  exaggerated  if  wc 
wish  to  insure  skill  or  even  sound  information  on 
this  subject. 

I.  A  square  number,  multiplied  by  a  square? 
number,  the  product  will  be  a  square  number. 

II.  A  square  number,  divided  by  a  square  num- 
ber, the  quotient  is  a  square. 

III.  A  cube  number,  multiplied  by  a  cube,  the 
product  is  a  cube. 

IV.  A  cube  number,  divided  by  a  cube,  the  quo- 
tient will  be  a  cube. 

V.  If  the  square  root  of  a  number  is  a  compos- 
ite number,  the  square  itself  may  he  divided  into 
integer  square  factors ;  but  if  the  root  is  a  prime, 
number,  the  square  can  not  be  separated  into  square 
factors  loitJwut  fractions. 

VI.  If  the  unit  figure  of  a  square  number  is  5,  we 
may  multiply  by  the  square  number  4,  and  we  shall 
have  another  square,  whose  unit  period  will  be 
ciphers. 

VII.  If  the  unit  figure  of  a  cube  is  5,  we  may 
multiply  by  the  cube  number  8,  and  produce  an 
othftr  cube,  whose  unit  period  will  be  ciphers. 

226 


SQUARE  AND  CUBE  ROOTS. 


227 


K.  B.  If  a  supposed  cube,  whose  unit  figure  is 
5,  be  multip-ied  by  8,  and  the  product  does  not  give 
three  ciphers  on  the  right,  the  number  is  not  a  cube. 

We  present  the  following  table,  for  the  pupil  to 
compare  the  natural  numbers  with  the  unit  Jigure 
of  th^ir  squares  and  cuhes^  that  he  may  be  able  to 
extract  roots  by  inspection. 


Numbers 

1 

2 

3 

4 

6 

6 

7 

8 

9 

10 

Bijiiares 

1 

4 

9 

16 

25 

36 

49 

64 

81 

100 

Gnbes 

1 

8 

27 

64 

126 

216 

343 

612 

729 

1000 

EXERCISES  FOR  PRACTICE. 

1.  What  is  the  square  root  of  625?     Ans.  25 
If  the  root  is  an  integer  number^  we  may  know, 

by  the  inspection  of  the  table,  that  it  must  be  25, 
as  the  greatest  square  in  6  is  2,  and  5  is  tho  only 
figure  whose  square  is  5  in  its  unit  place. 

Again,  take  625 

Multiply  by  4     4  being  a  sqi-are. 

2500 
The  square  root  of  this  product  is  obviously  50; 
but  this  must  be  divided  by  2,  the  square   root  of 
4,  which  gives  25,  the  root. 

2.  What  is  the  square  root  of  6561  ?     Ai%.  81. 
As  the  unit  figure^  in  tlis  example,  is  1,  and  Id 


228      ORTON  &  Sadler's  calculator. 

the  line  of  squares  in  the  table,  we  find  1  only  at  1 
and  81,  we  will,  therefore,  divide  6561  by  81,  and  we 
find  the  quotient  81 ;  81  is,  therefore,  the  square  root. 

3.  Whatisthe  square  root  of  106729?  Ans.  327 
As  the  unit  figure,  in  this  example,  is  9,  if  the 

number  is  a  square,  it  must  divide  by  either  9,  or 
49.  After  dividing  by  9  we  have  11881  for  the 
other  factor,  a  prime  number,  therefore  its  root  is  a 
prime  number=109.  109,  multiplied  by  3,  the 
root  of  9,  gives  327  for  the  answer. 

4.  What  is  the  root  of  451584?         Ans.  672. 
As  the  unit  figure  is  4,  and  in  the  line  of  squares 

we  find  4  only  at  4  and  64,  the  above  number,  if  a 
square,  must  divide  by  4,  or  64,  or  by  both. 

We  will  divide  it  by  4,  and  we  have  the  factors 
4  and  112896.  This  last  factor  closes  in  6  ;  there- 
fore, by  looking  at  the  table,  we  see  it  must  divide 
by  16,  or  36,  etc. 

We  divide  by  36,  and  we  have  the  factors  36  and 
3136;  divide  this  last  by  16,  and  we  have  16  and 
196 ;  divide  this  last  fraction  by  4,  and  we  have  4 
and  49. 

Take  now  our  divisors,  and  last  factor,  49,  and 
we  have  for  the  original  number  the  product  of 
4X36X16X4X49;  the  roots  of  which  are  2x6 
X 4X2X7,  the  products  of  which  are  672,  the 
tnswer 

5.  Extract  the  square  root  of  2025.     An$,  46. 


SQUARE   AND   CUBE   ROOTS.  229 

1st.  Divide  by  the  square  number  25,  and  wft 
find  the  two  factors,  25x81,  as  equivalent  to  the 
given  number.  Roots  of  thes«  factors,  byd=-:^4:b^ 
the  answer. 

Again,  multiply  by  the  square  number  4,  when 
a  number  ends  in  25,  and  we  have  8100,  root  90, 
half  of  which,  because  we  multiplied  by  4,  the 
equare  of  2,  is  45,  the  answer. 

Problems  on  the  Right-angled   Triangle, 

1.  The  top  of  a  castle  is  45  yards  high,  and  is 
surrounded  with  a  ditch,  60  yards  wide ;  required 
the  length  of  a  ladder  that  will  reach  from  the 
outside  of  the  ditch  to  the  top  of  the  castk. 

Ans.  75  yards. 

This  is  almost  invariably  done  by  squaring  45 
and  60,  adding  them  together,  and  extracting  the 
pquare  root ;  but  so  much  labor  is  never  necessary 
when  the  numbers  have  a  common  divisor^  or  whon 
the  side  sought  is  expressed  by  a  composite  number. 

Take  45  and  60 ;  both  may  be  divided  15,  add 
they  will  be  reduced   to  3  and  4.     Square  these, 
9  f  16=25.     The  square  root  of  25  is  5,  whi'^tv 
multiplied  by  15,  gives  75,  the  answer. 
Abbreviations  in  Cube  Root 

1.  Wha^.  is  the  cube  root  of  91125  ?     Am,  41 
Multiply  by  8 

729000 
20 


230      OETON  &  Sadler's  calculatopw 

New,  729  being  the  cube  of  9,  the  n»ot  of 
729000  is  90 ;  divide  this  by  2,  the  cube  root  of  8, 
and  we  have  45,  the  answer. 

When  it  is  requisite  to  multiply  several  numbers 
together  and  extract  the  cube  root  of  their  pro- 
duct, try  to  change  them  into  cube  factors  and  ex 
tract  the  root  before  multiplication. 

EXAMPLES. 

1.  What  is  the  side  of  a  cubical  miund  equai 
to  one  288  feet  long,  216  feet  broad,  anvl  48  feet 
high? 

The  common  way  of  doing  this,  is  to  multiply 
these  numbers  together  and  extract  the  root — a 
\engthy  operation.  But  observe  that  216  is  a  cube 
number,  and  288=2x12x12,  and  48==4xl2; 
therefore  the  whole  product  is216x8xl2Xl2X 
12.  Now,  the  cube  root  of  216  is  6,  of  8  is  2,  an<\ 
of  12'  is  12,  and  the  product  of  6x2x12=144, 
the  answer. 

2.  Required  the  cube  root  of  the  product  of 
448X392  the  short  way.  Ans.  5f^. 

We  can  extract  the  root  of  cube  numbers  by  in- 
ipectwn  when  they  do  not  contain  more  than  tw6 
periods. 


SQUARE   AND   CUBE    ROOTS.  23] 

RvLE. — As  there  will  he  two  figure9  in  the  root^  the  first 
may  easily  he  found  menially^  or  hy  the  Table  of  Powers  \  and 
\f  the  unit  figure  of  the  power  he  1,  the  unit  figure  in  the  rod 
will  he  1  ;  and  if  it  he  8,  the  root  will  be  two]  and  if  7,  it 
will  be  3 ;  and  if  the  unit  of  the  power  he  6,  ihe  unit  of  th 
'oot  will  be  6 ;  and  if  6,  it  v)ill  he  b\  j/"  3,  it  will  bel  \  \f% 
it  vnll  he  8 ;  and  if  the  unit  of  the  power  he  9,  the  unit  of  the 
foot  will  he  9.  This  will  appear  evident  hy  inspecting  iJu 
Table  of  Powers. 

EXAMPLES. 

Find  the  cube  root  of  195112.  Thia  numlver 
jonsists  of  two  periods.  Compare  the  superior 
period  with  the  cubes  in  the  table,  and  we  find  that 
195  lies  between  125  and  216.  The  cube  root  of 
the  tens,  then,  must  be  5.  The  unit  figure  of  the 
eriven  cube  is  2 ;  and  no  cube  in  the  table  has  2 
for  its  unit  figure,  except  512,  whose  root  is  8; 
therefore  58  is  the  root  required. 

What  is  the  cube  root  of  97336  ?        Ans  46. 

Explanation. — By  examining  the  left  hand 
period,  we  find  the  root  of  97  is  4,  and  the  cube  of 
i  is  64,  The  root  can  not  be  5,  because  the  cube 
of  5  is  125.  The  unit  figure  of  the  given  cube  is 
6 ,  and  no  cube  in  the  table  has  6  for  its  unit  fig- 
are,  except  216,  whose  root  is  6  ;  the  answer,  there- 
Fore,  is  46. 

The  number  912673  is  a  cube  ,  what  is  its  root? 

Am.  97. 

Observe,  the  root  of  the  superior  period   most 


232      ORTON  &  Sadler's  calculator. 

be  9,  and  the  root  of  the  unit  period  must  be  some 
number  which  will  give  3  for  its  unit  figure  when 
cubed ;  and  7  is  the  only  figure  that  will  answer. 

The  following  numbers  are  cubes ;  required  their 
roots 

1.  What  is  the  cube  root  of  59319?  Ans.  39 

2.  What  is  the  cube  root  of  79507  ?  Ans.  43. 
a  What  is  the  cube  root  of  117649?  Ans.  49, 
4.  What  is  the  cube  root  of  110592?  Ans.  48. 
6.  What  is  the  cube  root  of  357911  ?  Ans.  71. 

6.  What  is  the  cube  root  of  389017  ?    Ans.  73. 

7.  What  is  the  cube  root  of  571787  ?  Ans.  83. 
When  a  cube  has  more  than  two  periods,  it  can 

generally  be  reduced  to  two  by  dividing  by  some 
one  or  more  of  the  cube  numbers,  unless  the  root 
is  a  prime  number. 

The  number  4741 632  is  a  cube ;  required  its 
root.  Here  we  observe  that  the  unit  figure  is  2  ; 
the  unit  figure  of  the  root  must  therefore  bo  the 
root  of  512,  as  that  is  the  only  cube  of  the  9  dig- 
its whose  unit  figure  is  2.  The  cube  root  of  512 
JB  8 ;  therefore  8  is  the  unit  figure  in  the  root,  and 
the  root  is  an  even  number,  and  can  be  divided  by 

2 ,  and  of  course  the  cube  itself  can  be  divided  by 

3,  the  cube  of  2.  8)4741632 


592704 
Now  as  the  first  number  was  a  cube,  and  being 


SQUARE   AND   CUBE   ROOTS.  233 

divided  by  a  cube,  the  number  592704  must  be  a 
cube,  and  by  inspection,  as  previously  explained, 
its  root  must  be  84,  which,  multiplied  by  2,  gives 
168,  the  root  required. 

The  number  13312053  is  a  cube;  what  is  its 
root?  Ans.  237. 

As  there  are  three  periods,  there  must  be  three 
figures,  units,  tens,  and  hundreds,  in  the  root;  the 
hundreds  must  be  2,  the  units  must  be  7.  Let  us 
then  divide  the  2d  figure,  or  the  tens,  in  the  usual 
wat/j  and  we  have  237  for  the  root. 

Again,  divide  13312053  by  27,  and  we  have 
493039  for  another  factor.  The  root  of  this  last 
number  must  be  79,  which,  multiplied  by  3,  the 
cube  root  of  27,  gives  237,  as  before. 

The  number  18609625  is  a  cube;  what  is  its 
root? 

As  this  cube  ends  with  5,  we  will  multiply  it 
by  8: 

18609625 
8 


148877000 
As  the  first  is  a  cube,  this  product  must  be  a  cube; 
and  as  far  as  labor  is  concerned,  it  is  the  same  ae 
reduced  to  two  periods,  and  the  root,  we  perceive 
at  once,  must  be  530,  which,  divided  by  2,  give* 
265  for  the  root  required. 


234      ORTON  &  Sadler's  calculator. 

S.  B. — If  a  number,  whose  unit  figure  is  5,  be 
multiplied  by  8,  and  does  not  result  in  threv  cipJient 
on  the  right,  the  number  is  not  a  cube. 

To  find  the  Approximate  Cube  Root  o/  Surds, 

E.ULE. —  Take  the  nearest  rational  cube  to  the  given 
numheTy  and  call  it  the  assumed  cube  ;  or  assume  a 
root  to  the  given  number  and  cube  it.  Double  the 
assumed  cube  and  add  the  number  to  it;  also  double 
the  number  and  add  the  assumed  cube  to  it.  Take 
the  difference  of  these  sums^  then  say^  As  double  of 
the  assumed  cube,  added  to  the  number,  is  to  this  dif- 
ference, so  is  the  assumed  root  to  a  correction. 

This  correction,  added  to  or  subtracted  from  the 
assumed  root,  as  the  case  may  require,  will  give  the 
oube  root  very  nearly. 

By  repeating  the  operation  with  the  root  last 
found  as  an  assumed  root,  we  may  obtain  results 
to  any  degree  of  exactness ;  one  operation,  how- 
ever, is  generally  sufficient. 

EXAMPLES. 

1.  Kequired  the  cube  root  of  66. 

The  cube  root  of  64  is  -i.  Now  it  is  manifest 
that  the  cube  root  of  Q^  is  a  littlo  more  than  4, 
and  by  taking  a  similar  proportion  to  the  preced- 
ing, we  have 

64x2«128     2y66=132 
66  64 

194  196:  :4:  to  root  of- 6a 


SQUARE  AND  CUBE  ROOTS. 

Or,  194  :  2  :  :  4  :  to  a  correction 

194)8.0000(0.04124 
7  76 

240 
194 

460 

388 


235 


720 
Therefore  tho  cube  root  of  66  is  4.04124. 
2.  Required  the  cube  root  of  123. 
Suppose  it  5  ;  cube  it,  aud  we  have  125. 
Now  we  perceive  that  the  cube  of  5  being  greatei 
ihaD  123,  the  correction  for  5  must  be  subtracted 

2X125=250     246 
Add  123     125 


As 
Or, 


373 


373  :  371  : 
2     :  :     5 


:  5  :  root  of  123. 
:     correction  for  5 


373)10*0000(0.02681 
7  46 

2  540 
2  238 


From  5.00000 
Take  0.02681 


3020 

2984 


Am.    4.97319 


360. 


236      ORTON  &  Sadler's  calculator. 

3.  What  is  the  cube  root  of  28  ?     Am.  3,U3658-|. 

4.  What  is  the  cube  root  of  26  ?     Ans,  2,96249+ 

5.  What  is  the  cube  root  of  214?  Ans.  5,.98142  + 

6.  What  is  the  cube  root  of  346  ?  Ans.  9,02034+ 
The  above  being  very  near  integral  cubes—  thai 

tA.  28  and  26  are  both  near  the  cube  number  27, 
214  is  near  216,  etc.  All  numbers  very  near  cube 
bumbers  are  easi/  of  solution. 

We  now  give  other  examples,  more  distant  from 
integral  cubes,  to  show  that  the  labor  must  be  more 
lengthy  and  tedious,  though  the  operation  is  the 
same. 

1.  What  is  the  cube  root  of  3214?  Ans.  14,75758. 

Suppose  the  root  is  15 — its  cube  is  3375,  which, 
being  greater  than  3214,  shows  that  15  is  too  great ; 
the  correction  will  therefore  be  subtractive. 

By  the  rule,  9964  :  161  :  :  15.  0,243,  the 
correction. 

Assumed  root 15,0000 

Less 2423 

Root  nearly 14,7577 

Now  assume  14,7  for  the  root,  and  go  over  the 
operation  again,  and  you  will  have  the  true  root  to 
8  or  10  places  of  decimals. 

N.  B. — Roots  of  component  powers  may  be  ob- 
tained more  readily  thus : 

For  the  4th  root,  take  the  square  root  of  the 
gquare  root. 


MEiNSURATlON  OR  PRACTICAL  GEOMETEY, 

MEASUREMENT   OF   GRINDSTONES. 

Grindstones  are  sold  by  the  stone,  and  their  con- 
tents  found  as  follows:* 

lluLE. —  To  the  whole  diameter  add  half  of  the 
diameter^  and  multiply  the  sum  of  these  by  the 
same  half  and  this  product  by  the  thickness ;  di- 
vide this  last  number  by  1728,  and  the  quotient  is 
the  contents,  or  answer  required, 

EXAMPLES. 
What  are  the  contents  of  a  grindstone  24 
Inches  diameter,  and  4  inches  thick 


24+12x12X4 

=1  stone.  Am. 

1728 
2.  What  are  the  contents  of  a  grindstone  36 
inohes  diameter,  and  4  inches  thick.    Am.  2J  stone. 

Mensuration  of  Superficies  and  Solids, 
Superficial  measure  is  that  which  relates  to  length 
And  breadth  only,  not  regarding  thickness.  It  i« 
made  up  of  squares,  either  greater  or  less,  accord- 
ing to  the  different  measures  by  which  the  dimen« 
sions  of  the  figure  are  taken  or  measured.  Land 
10  measured  by  this  measure,  its  dimensions  being 

*24  inches  iu  diameter,  and  4  inches  thick  make  a  ston* 

237 


238      ORTON  &  Sadler's  calculator. 

usuallj  taken  in  acres,  rods,  and  links.  The  cod 
tents  of  boards,  also,  are  found  by  this  measure, 
their  dimensions  being  taken  in  feet  and  inches. 
Because  12  inches  in  length  make  1  foot  of  long 
measure,  therefore  12x12=144,  the  square  inches 
in  a  superficial  foot,  etc. 

Note. — Superficial  means  lying  on  the  surface 

To  find  the  area  of  a  square  having  equal  sides. 

Rule. — Multiply  the  side  of  the  square  into 
itself  and  the  product  will  be  the  area,  or  superfi- 
cial content  of  the  same  name  with  the  denomina- 
tion taken,  whether  inches,  feet,  yards,  rods,  and 
links,  or  acres. 

EXAMPLES. 

1.  How  many  square  feet  of  boards  are  contain- 
ed in  the  floor  of  a  room  which  is  20  feet  square? 

20X20=400  feet,  the  answer. 

2.  Suppose  a  square  lot  of  land  measures  36 
rods  on  each  side,  how  many  acres  does  it  contain  ? 

36X36«1296  square  rods.     And 
1296-T-160=:8  acres,  16  rods,  Ans. 

As  160  square  rods  make  an  acre,  therefore  we  di- 
wde  1296  by  IGO  to  reduce  rods  to  acres. 

N.  B. — The  shortest  way  to  work  this  example 
is,  to  cancel  36x36  with  the  divisor  160.  Arrange 
the  example  as  below ;  (divide  both  terms  by  4x4:) 

36X36  9X9 

same  as =8.1  acres,  or  Sao.  16  rods. 

160  10 


MENSURATION    OR   PRACTICAL   GEOMETRY.  239 

To  measure  a  parallelogram  or  long  square. 
Rule. — Multiply  the  length  by  the  breadth^  and 
the  product  icill  be  the  area^  or  superficial  content^ 
in  the  same  name  as  that  in  which  the  dimension 
was  taken^  whether  inches^  feet^  or  rods^  etc, 

EXAMPLES 

1.  A  certain  garden,  in  form  of  a  long  square,  is 
96  feet  long,  and  54  feet  wide  ;  how  many  square 
feet  of  ground  are  contained  in  it  ? 

Ans.  96X54=5184  square  feet. 

2.  A  lot  of  land,  in  form  of  a  long  square,  is 
120  rods  in  length,  and  60  rods  wide;  how  many 
acres  are  in  it?  120x60=7200  sq.  rods.  And 
7200-^160=45  acres,  Ans, 

Note. — The  learner  must  recollect  that  feet  in 
length,  multipled  by  feet  in  breadth,  produce  square 
feet ;  and  the  same  of  the  other  denominations  of 
lineal  measure. 

Note. — Both  the  length  and  breadth,  if  not  in 
units  of  the  same  denomination,  must  be  made  ko 
before  multiplying. 

3.  How  many  acres  aro  in  a  field  of  oblong 
form,  whose  length  is  14,5  chains,  and  breath  9,75 
chains?  Ans.  14ac.  Orood,  22rods- 

NOTE. — The  Gunter's  chain  is  66  feet,  or  4  rods, 
long,  and  contains  100  links.  Therefore  if  dimen- 
tions  be  given  in  chains  and  decimals,  point  afl 
^om  the  product  one  more  decimal  place  than  arc 


240      ORTON  &  Sadler's  calculator. 

eontained  m  both  factors,  and  it  will  be  acres  and 
decimals  of  an  acre  ;  if  in  chains  and  links,  do  the 
same,  because  links  are  hundredths  of  chains,  and 
therefore  the  same  as  decimals  of  them.  Or,  as  1 
chain  wide,  and  10  chains  long,  or  10  square  chains, 
or  100000  square  links,  make  an  acre,  it  is  the  same 
as  if  you  divide  the  links  in  the  area  by  I'OOOOO. 

4.  If  a  board  be  21  feet  long  and  18  inches 
broad,  how  many  square  feet  are  contained  in  it: 

18  inches=:l,5  foot;  and  21 X  1,5=31,5  ft.  Am. 

Or,  in  measuring  boards,  you  may  multiply  the 
length  in  feet  by  the  breadth  in  inches,  and  divide 
the  product  by  1 2  ;  the  quotient  will  give  the  an- 
swer in  square  feet,  etc.  21  xl8 

Thus,  in  the  last  example,  =31^sq.  ft., 

as  before.  12 

5.  If  a  board  be  8  inches  wide,  how  much  in 
length  will  make  a  foot  square  ? 

Rule. — Divide  144  by  the  width ;  thus^  8)144 

Am,  18  en. 

6.  If  a  piece  of  land  be  5  rods  wide,  how  many 
rods  in  length  will  make  an  acre  ? 

Rule. — Divide  160  by  the  widths  and  the  qua 
Uent  will  be  the  length  required;  thus, 
6)160 

Ans,  32  rods  in  length. 


MENSURATION    OR    PRACTICAL    GEOMETRY.  241 

Note. — Wlien  a  board,  or  any  other  surfj*ctf»,  i« 
wider  at  one  end  than  the  other,  but  yet  is  of  a 
true  taper,  you  may  take  the  breadth  in  the  middle, 
or  add  the  widths  of  both  ends  together,  and  halve 
the  sum  for  the  mean  width ;  then  multiply  the 
said  mean  breadth  in  either  case  by  the  length , 
the  product  is  the  answer  or  area  sought. 

7.  How  many  square  feet  in  a  board,  10  feei 
long  and  13  inches  wide  at  one  end,  and  9  inches 
wide  at  the  other?         13+9 

1=11  in.,  mean  width. 

2       ft.     in. 

10X11 

=:9^ft.,  Ans, 

12 

8.  How  many  acres  are  in  a  lot  of  land  which  is. 
40  rods  long,  and  30  rofls  wide  at  one  end,  and  20 
fod?  w^de  at  th^  other? 

30+20 

=25  rods,  mean  width. 

2  Then,  25x40 

=6 J  acres,  Ans, 

160 
9    If  a  farm  lie  250  rods  on  the  road,  and  at  odo 
3nd  be  75  rods  wide,  and  at  the  other  55  rods  wide, 
bow  many  acres  does  it  contain  ? 

Ans.  101  acres,  2  roods,  10  rods. 

N.  B. — Always  arrange  your  example  as  above, 
and  cancel  the  factors  common  to  both  terms  before 
maltiplying 


242   ORTON  &  Sadler's  calculator. 

Case  3. — To  measure  the  surface  of  a  triangle, 

Pefinition. — A  triangle  is  any  three-cornered 

figure  which  is  bounded  by  three  right  lines* 

Rule. — Multiply  the  hose  of  the  given  triangle 

into  half  its  perpendicular  hight,  or  half  the  base 

into  the  whole  perpendicular,  and  the  product  will 

be  the  area. 

EXAMPLES. 

1.  Required  the  area  of  a  triangle  whose  base  or 
longest  side  is  32  inches,  and  the  perpendicular 
bight  14  inches. 

14-f-2=7=half  the  perpendicular.     And 
32x7=224sq.  in.,  Ans. 

2.  There  is  a  triangular  or  three-cornered  lot  of 
land  whose  base  or  longest  side  is  51 J  rods;  the 
perpendicular,  from  the  corner  opposite  to  the  base, 
measures  44  rods ;  how  many  acres  does  it  contain  ? 

44-i-2=22=half  the  perpendicular. 
And  51,5x22 

=7  acres,  13  rods,  Ans, 

160 

Joists  and  planks  are  measured  by  the  following: 

Rule. — Find  the  area  of  one  side  of  the  joist  or 

plank  by  one  of  the  preceding  rules ;  then  multiply 

it  by  the  thickness  in  inches^  and  the  last  product 

will  be  the  superficial  content, 

*  A  triangle  may  be  <»ither  right-angled  or  oblique. 


MENSURATION   OR    PRACTICAL   GEOMETRY.  243 

EXAMPLES. 

1.  Wnat  is  the  area,  or  superficial  content,  or 
board  measure,  of  a  joist,  20  feet  long,  4  inches 
wide,  and  3  incLes  thick  ?     20x4 

X3=20ft.,  ^n«. 

12 

2.  If  a  plank  be  32  feet  long,  17  inches  wide, 
and  3  inches  thick,  what  is  the  board  measure  of 
it?  Ans,  136  feet 

Note. — There  are  some  numbers,  the  sum  of 
whose  squares  makes  a  perfect  square ;  such  are  3 
and  4,  the  sum  of  wk>se  squares  is  25,  the  square 
root  of  which  is  5 ;  consequently,  when  one  leg 
of  a  right-angled  triangle  is  3,  and  the  other  4, 
the  hypotenuse  must  be  5.  And  if  3,  4,  and  5,  be 
multiplied  by  any  other  numbers,  each  by  the  same, 
the  products  will  be  sides  of  true  right-angled  tri- 
angles. Multiplying  them  by  2,  gives  6,  8,  and  10, 
by  3,  gives  9,  12,  and  15 ;  by  4,  gives  12,  16,  and 
20,  etc.;  all  which  are  sides  of  right-angled  tri- 
angles. Hence  architects,  in  setting  off  the  corners 
of  buildings,  commonly  measure  6  feet  on  one  side, 
and  8  feet  on  the  other  ;  then,  laying  a  10-foot  pole 
across  from  those  two  points,  it  makes  the  corner 
a  true  right-angle. 

N.  B. — The  solutions  of  the  foregoing  problems 
are  all  very  brief  by  canceling. 


244      ORTON  &  Sadler's  calculator, 

7b  find  the  area  of  any  triangle  when  the  three  ndes 
only  are  given. 
Rule. — From  half  the  sum  of  the  three  sides  sub- 
tract each  side  severally;  multiply  these  three  re- 
mainders and  the  said  half  sum  continually  together  ; 
^hen  the  square  root  of  the  last  product  will  he  the 
area  of  the  triangle, 

EXAMPLE. 
Suppose   I  have  a  triangular  fish-pond,  whose 
three  sides  measure  400,  348,  and  312yds;   what 
quantity  of  ground  does  it  cover  ? 

Ans.  10  acres,  3  roods,  S-frods. 

Note. — If  a  stick  of  timber  be  hewn  three 
square,  and  be  equal  from  end  to  end,  you  find 
the  area  of  the  base,  as  in  the  last  question,  in 
inches ;  multiply  that  area  by  the  whole  length, 
and  divide  the  product  by  144,  to  obtain  the  bolid 
content* 

If  a  stick  of  timber  be  hewn  three  square,  be  12 
feet  long,  and  each  side  of  the  base  10  inches,  the 
perpendicular  of  the  base  being  8f  inches,  what  ia 
its  solidity?  Ans.  3,6-[-feet. 

PROBLEM  1. 

The  diameter  given,  to  find  the  circumference. 

Rule. — As  7  are  to  22,  so  is  the  given  diameter 
t-o  the  circumference ;  or^  more  exactly^  o^  113  are 
to  355,  so  is  the  diameter  to  the  circumference,  eta 


MENSURATION    OR   PRACTICAL   GEOMETRY.  245 

EXAMPLES. 

1 .  What  is  the  circumference  of  a  wheel,  whose 
diameter  is  4  feet? 

As  7  :  22  :  :  4  :  12,57+ft.,  the  cir-um.,  Am, 

2.  What  is  the  circumference  of  a  circle,  whose 
liameter  is  35  rods  ? 

As  7  :  22  :  :  35  :  110  rods,  Am, 

Note. — To  find  the  diameter  when  the  circum- 
ference is  given,  reverse  the  foregoing  rule,  and  say, 
a«  22  are  to  7,  so  is  the  given  circumference  to  the 
required  diameter;  or,  as  355  are  to  113,  so  is  the 
circumference  to  the  diameter. 

3.  What  is  the  diameter  of  a  circle,  whose  cir- 
cumference is  110  rods? 

As  22  :  7  :  :  110  :  35  rods,  the  diam.,    Ans, 

Case  5. — To  find  how  many  solid  fed  a  round 
stick  of  timber,  equally  thick  from  end  to  end^ 
will  contain^  when  hewn  square. 
Rule. — Multiply  twice  the  square  of  its  serm-dv- 
ameler,  in  inches,  by  the  length  in  the  feet;  then  divide 
the  product  by  144,  and  the  quotient  will  be  the  an- 
swer. 

N  B. — When  multiplication  and  division  are 
combined,  always  cancel  like  factors.  When  the 
numbers  are  properly  arranged,  a  few  clips  with 
the  pencil,  and,  perhaps,  a  irijling  multiplication 
will  suffictj. 


246      ORTON  &  Sadler's  calculator. 

For  the  practical  convenience  of  those  who  have  occasion  to 
refer  to  mensuration,  we  have  arranged  the  following  useful  -able 
3f  multiples.  It  covers  the  whole  ground  of  practical  geometry,, 
and  should  be  studied  carefully  by  those  who  wish  to  be  skilled  in 
this  beautiful  branch  of  matliematics : 

TABLE   OF   MULTIPLES. 

Diameter  of  a  circle  X  ^-1416  —  Circumference. 

Radius  of  a  circle  X  6.283186  —  Circumference. 

Square  of  the  radius  of  a  circle  X  3.1416  —  Area. 

Square  of  the  diameter  of  a  circle  X  0.7854  =-  Area. 

Square  of  the  circumference  of  a  circle  X  0-07958  =  Area. 

Half  the  circumference  of  a  circle  X  by  half  its  diameter  -«  Ai^ea. 

Circumference  of  a  circle  X  0.159165  =-  Radius. 

Square  root  of  the  area  of  a  circle  X  0.66419  «-  Radius. 

Circumference  of  a  circle  X  0.31831  —  Diameter. 

Square  root  of  the  area  of  a  circle  X  112838  =«  Diameter. 

Diameter  of  a  circle  X  0.86  -=  Side  of  inscribed  equilateral  trianglo. 

Diameter  of  a  circle  X  0.7071  —  Side  of  an  inscribed  square. 

Circumference  of  a  circle  X  0.225  —  S\de  of  an  inscribed  square. 

Circumference  of  a  circle  X  0.282  —  Side  of  an  equal  square. 

Diameter  of  a  circle  X  0.88C2  —  Side  of  an  equal  square. 

Base  of  a  triangle  X  t»y  K  *^®  altitude  —  Area. 

Multiplying  both  diameters  and  .7854  together  =  Area  of  an  elllpso. 

Surface  of  a  sphere  X  by  3^  of  its  diameter  =«  Solidity. 

Circumference  of  a  sphere  X  by  its  diameter  —  Surface. 

Square  of  the  diameter  of  a  sphere  X  3.1416  —  Surface. 

Square  of  the  circumference  of  a  sphere  X  0.3183  «  Surface. 

Cube  of  the  diameter  of  a  sphere  X  0.5236  —  Solidity. 

Cube  of  the  radius  of  a  sphere  X  41888  —  Solidity. 

Cube  of  the  circumference  of  a  sphere  X  0.016887  —  Solidity. 

Square  root  of  the  surface  of  a  sphere  X  0.56419  —  Diameter, 

Square  root  of  the  surface  of  a  sphere  X  1.772454  —  Circumference. 

Cube  root  of  the  solidity  of  a  sphere  X  1-2407  —  Diameter. 

Cube  root  of  tlie  solidity  of  a  sphere  X  3.8978  —  Circumference, 

Radius  of  a  sphere  X  1.1547—  Side  of  inscribed  cube. 

Square  root  of  {%  of  the  square  of)  the  diameter  of  a  sphere  — 

Side  of  inscribed  cube. 
Area  of  its  base  X  by  >«J  of  its  altitude  —  Solidity  of  a  cone  or  pyr- 

arrid,  whether  round,  squaie,  or  triangular. 
irea  if  one  of  its  sides  X  6  >=-  Surface  of  a  cube. 
AHitude  of  Tapezoid  X  %  the  sum  ct  its  parallel  sides  —  Area. 


WEIGHTS   AND   MEASURES. 


247 


AVOIRDUPOIS  WEIGHT 
Is  the  standard  weight  for 
weighing  the  greater  por- 
tion of  articles  used  in 
trade  and  commerce,  such 
asgroceries,produce,  iron, 
coal,  hay,  cotton,  etc. 

TABLE. 
437V^  grains  ((;rr).l  ounce — ox. 

IG  oz 1  pound 76. 

25  lb 1  quiirter. .  .qr. 

4  qr 1  \\\\.\\''hX,cwt. 


EQUIVALENTS. 

T. 
1  = 

ciot. 
20  = 
1  = 

qr. 

80 
4 
1 

lb.                   02. 

~   2000  =  32000 

=   100  =   1600 

=   25  =   400 

1  =    16 

1 

=  14000000 
=   700000 
=   175000 
=     7000 
^     437i 

Scale  of  units  : — 437J,  10,  25,  4,  20. 

The  dram  is  now  seldom  used,  except  with  silk  manu- 
facturers.    The  ounce  is  divided  into  k  and  ^. 

The  following  denominations  are  also  used; 
LONG   OB    IRON    TON. 

28  lbs 1  quarter. 

4  qr.,  or  112  lbs 1  hundredweight. 

20  cwt.,  or  2240  lbs 1  ton. 

This  measurement  is  nearly  obsolete.  It  is  allowed  at 
the  Custom  House  in  estimating  duties,  and  in  whole- 
Bale  transactions  of  iron  and  coal. 

Note. — The  grain  avolnlniwis,  though  never  used,  is  tlie  same  as  the 
grain  in  Troy  weight.  7(Kio  gmina  make  the  avuirUuiKjis  pound,  and 
5700  grains  the  Truy  pound. 

IRON,  LEAD,   Etc. 

14  lbs 1  stone. 

21i  stone 1  pig. 

8  pigs 1  fother. 


248 


ORTON   &   SADLER  S    CALCULATOR. 


AVOIKDUPOIS  AT^-EIGHT. 

Miscellaneous  Table. 

14  pounds  of  Iron  or  Lead 1  stone. 

100        "       ''  Grain  or  Flour 1  cental. 

loo        "      "  Raisins 1  cask. 

100        "      "  Dry  Fish 1  quintal. 

100        "      "Nails 1  keg. 

196        "      "Flour 1  barrel. 

200        "      "  Pork,  Beef,  or  Fish 1  barreL 

240        "       "Lime 1  cask. 

280        "      "  Salt 1  barrel  of  Salt  at 

the  N.  Y.  Salt  Works. 


APOTHECAKIES' 
WEIGHT 

Is  used  by  Apothecaries 
and  Physicians  in  dis- 
pensing medicines,  not 
liquid.  The  grains  men- 
tioned in  the  following 
table  are  Troy. 


TABLE. 

20  Grains  [gr.  xx.) 1  scruple,  9 

3  Scruplos  O  iij) 1  dram,      g 

8  Drams  (.5  viij) 1  ounce,     ^ 

12  Ounces  (5  xij) 1  pound,    lb 

EQUIVALENTS. 

ft>         .?  5  9  gr. 

1     z:==     12     =     96     =     288     =     57G0 

1     =      8     ==       24     =      480 

1     ==        3     =        60 

1     =        20 

ScALR  OF  Units  :— 20,  3,  8,  12. 

The  only  difference  between  Troy  and  Apothecaries* 
weight  is  the  division  of  the  ounce.  The  pound,  ounce, 
and  grain  are  the  same.  Drugs  and  medicines  are  bought 
and  sold  in  quantities  by  Avoirdupois  weight; 


WEIGHTS  AND  MEASURES. 


249 


APOTHECARIES'  FLUID   MEASURE, 
Used  ill  mixing  liquid  medicines  by  measure. 

60  Minims  (m) 1  fluid  drachm,/,^ 

8  /.^ 1  fluid  ounce,    /g 

16/5 1  pint,      0.  ( Oct  arms.) 

8  0 1  gallon,  ( Coiig.  Congius.) 

EQUIVALENTS. 

128 


Cong. 
1     = 


0. 

8    = 
1     ^ 


16 
1 


/5 

1024 

128 

8 

1 


m 
61440 
7680 
480 
60 


Scale  of  Units:— 60,  8,  16,  8. 

Note. — One  fluid  ounce ^455.0944  Troy  grains. 
The  minim  is  a  droi)  of  pure  water,  and  is  equal  to  alx)ut  i^^^j  af  < 
grain  Troy. 

An  ordinary  teacupful  is  about  4  fluid  ounces. 
Common  tablespoonful  ^  a  fluid  ounce. 
Teaspoon  contains  about  45  drops. 

A^OOD    MEASURE, 
For    measuring    wood, 
rough  stone,  fences,  etc. 

16  Cu.  Ft 1  cord  ft. 

8  Cord  Feet  or 
128  (  u  >ie  Ft..l  cord. 

243  Cubic  Feet 1  ]>erch 

of  stone  or  masonry. 

A  cord  of  wood  is  a 
pile  8  feet  long,  by  4  feet 
wide,  and  4  feet  high. 

A  cord  foot  is  one  foot 
of  the  running  pile,  or  h 
of  a  cord. 

A  ])erch  of  stone  or 
masonry  is  16i  feet  long,  U  feet  wide,  and  1  foot  high. 

Wood.  Measuring  wood  in  the  load.  If  the  rack  is 
narrower  nt  the  bottom  than  at  the  top,  the  width  of  the 
logd  should  be  measured  half-way  from  base  to  top;  this 
will  ijive  the  average  width. 


250      OETON  &  Sadler's  calculator. 


DKY   MEASURE 

Is  used  in  measuring  arti- 
cles not  fluid,  such  as 
grains,  seeds,  vegetables, 
fruit,  salt,  etc. 

TABLE. 
2  ^inis{pt.).l  quart ....  qt, 

8  qt 1  peck.....pk, 

4pk 1  bushel....6w. 

36  bu lcli'ldron,cA, 


EQUIVALENTS. 


ch. 
1     = 


bu.  pk. 

36    =    144 

1     =        4 

1 


qt, 

1152 

32 

8 

1 


pL 

2304 

64 

16 

2 


Scale  of  units :— 2,  8,  4,  36. 

Note. — 1  gal.  Wine  Measure  contains  231  cu.  in.,  1  gal.  Ale  or  Beer 
Measure  (nearly  obsolete),  282  cu.  in.,  and  1  bu.  2150-j^^j^y  cu.  in. 

The  legal  bushel  of  the  United  States  is  the  old  Win- 
chester measure,  cylindrical  in  form,  18i  inches  in  diame- 
ter and  8  inches  deep,  containing  2150^*  ^^  cubic  inches. 
The  Imperial  bushel  of  England  is  2218^'jjYo  cubic  inches. 
32  English  bushels  equal  33  of  the  United  States. 

Heaped  Pleasure  is  the  contents  of  bushel  heaped  in  the 
shape  of  a  cone.  Corn  in  the  ear,  large  fruits,  vegetables, 
and  bulky  articles  are  sold  by  this  measure. 

Stricken  Measure  is  the  bushel  even  full,  having  been 
stricken  off  by  a  rule  or  striker.  Grain,  seeds,  etc.,  are 
sold  by  this  measure.  It  is  customary  to  allow  5  stricken 
measures  for  4  heaped  ones.  It  is  usual  to  quote  the  price 
of  grain,  etc.,  by  the  bushel,  but  more  frequently  to  deter- 
njiue  their  value  by  weight. 


WEIGHTS   AND   MEASURES. 


251 


CUBIC  OR  SOLID  MEASURE, 
Used  iu  measuring  any- 
thing containing  length, 
breadth,  depth,  and  thick- 
ness, such  as  timber, 
wood,  stone,  boxes,  stor- 
age capacity  of  rooms, 
bins,  cisterns,  etc. 

EQUIVALENTS. 

cu,ft.  cu.  in, 

1        =:        1728 

Scale  of  units  ;— 1, 1728.<J 

TABLE. 

1728  Cubic  Inches  {cu.  in.) 1  cubic  foot,  cto.ft. 

27  Cubic  Feet 1  cubic  yard,  cu.  yd. 

40  Cubic  Feet 1  ton  of  ship  cargo,     , 

50  Cu.  Ft.  of  Square  Timber..l  ton. 
One  cubic  yard  contains  46,656  cubic  inches. 
A  Registered  ton,  in  computing  the  tonnage  of  ships 
and  vessels,  is  100  cubic  feet  of  internal  capacity. 

In  measuring  cargoes,  a  ton  is  40  cubic  feet  in  the  United 
States,  and  40  cubic  feet  in  England. 

Light  articles  of  freight  are  generally  estimated  by  the 
space  occupied,  but  heavy  articles  by  weight. 

A  CUBIC  FOOT  OF 


Pounds, 
weighs  124 
127 


Common  soil 
Strong       " 
Loose  earth  or  sand  "        95 
Clay  "       135 

Lead  "       708J 

Brass  *'       534J 

Copper  *'       555 

Wrought  iron  "       486J 

Anthracite  coal       **   50-55 
Bituminous  '*  "  45-55 

Charcoal  (hard  w'd)"        18i 


Pounds. 
Clay  and  stones  weigh  100 
Cork  weicrhs    15 

Tallow  **       59 

Bricks  "      125 

Marble  "      171 

Granite  "      165 

Sea-water  "       64^0 

Oak  wood  "        55 

Red  pine  "        42 

White  pine  "        30 

Charc'l(  pine  wood)"     18 


MEASUREMENTS  AND   ESTBIATES. 

A  cubic  yard  of  earth  is  called  a  load. 

J?r/cA-*  are  of  various  dimensions.  The  average  size  :8 
8  inches  long,  4  inches  wide,  2  inches  thick.  27  bricks 
make  a  cubic  foot,  when  laid  dry.  Laid  in  mortar  k  to 
ji^  is  allowed  for  mortar.  Baltimore  and  Milwaukee 
bricks  are  8i  X  4J  X  2i  inches. 

Brick-  Work  is  generally  estimated  by  the  1000.  When 
measured  by  square  measure  the  work  is  understood  to 
be  12  inches  thick. 

Board  and  Lumber  Measure.  All  estimates  are  made 
on  one  inch  in  thickness;  for  every  \  inch  in  thickness  k 
price  is  added. 

Board  feet  are  changed  to  cubic  feet  by  dividing  by  12. 

Cubic  feet  to  board,  by  multiplying  by  12. 

Estimating  "Work  by  Artificers. 

In  material  only  is  allowance  made  for  windows,  doors 
and  cornices.     No  allowance  being  made  in  estimating  the 
work.    The  size  of  a  cellar  or  wall  is  estimated  by  the 
measurement  of  the  outside.     No  allowance  for  corners. 
Estimates,  How  Made. 

By  the  square  foot,  as  in  glazing,  stone-cutting,  etc. 

By  the  square  yard,  as  in  plastering,  painting,  etc. 

By  the  square  (100  sq.  ft.),  as  in  flooring,  rooting,  slating, 
paving,  etc. 

Painting  of  mouldings,  cornices,  etc.,  the  estimate  is  by 
measuring  the  entire  surface. 

SIZE  OF  WAILS. 

2-penny 1    inch 557  nails  per  pound. 

4-penny 1^  inches 353     *  * 

5-])eniiy 1|        "     232  **  '* 

6-i)enny  2         "     167  "  ** 

7-penny 2i        "     141  "  " 

8-penny 2^        "     101  **  ** 

10-penny 21        "     (j8  **  ** 

12-penny 3          "     54  '*  ** 

20-i>ei"iv 3i       "     34  '*  " 

bjukes 4          "     16  "  " 

Si>ikes 4i        "     12  "  " 

Spikes 5         "     10  " 

From  this  table  au  estimate  of  quantity  aud  suitable  ti/ts  Icr  on^ 
Job  of  work  cau  be  made. 

252 


WEIGHTS   AND   MEASURES. 


253 


LIQUID  OR  RATINE  MEASURE, 
Used  in  measuring 
liquids,  such  as  liquors, 
vinegar,  molasses,  oils, 
etc.,  and  estimating  the 
capacity  of  vessels  de- 
signed to  contain  them. 

TABLE. 

4  gills  {gi.)  1  pint,  pt. 

2   pints 1  quart,  qt. 

4   quarts ....1  gallon,  gal, 
3H  gallons.. .1  barrel,  bbl. 
2   barrels.. .1  hogs'd,  hhd,  ' 


hhd. 
1     = 


bbL 
2 
1 


EQUIVALENTS. 
gal. 
63 


3U 

1 


qt 

252 

126 

4 

1 


pt 

504 

252 

8 

2 

1 


gi. 
^    2016 
=    1008 
=        32 
=^  8 

=  4 


Scale  of  Units:— 4,  2,  4,  31  J,  2. 

Note.— The  gallon  must  contain  exactly  10  pounds  Avoirdupois,  of 
pure  water,  at  a  teinperatnro  of  62°,  the  barometer  being  at  30  inches. 
It  is  the  standard  unit  of  measure  of  capacity  for  liquids  and  dry  goods 
of  every  description,  and  is  y  larger  than  the  old  wine  measure,  tj^j 
larger  than  the  old  dry  measure,  and  -^^  less  than  the  old  ale  measure. 
The  wine  gallon  must  contain  231  cubic  inches. 

Barrels  used  in  commerce  are  made  of  various  sizes, 
from  30  to  45,  and  even  56  gallons.  There  is  no  definite 
measure  called  a  hogshead,  they  are  usually  gauged,  and 
have  their  capacities  in  gallons  marked  on  them.  The 
Standard  gallon,  United  States,  contains  231  cubic  inches. 
The  Imperial  gallon,  Great  Britain,  277.274  cubic  inches, 
and  is  equal  to  j  more  than  the  United  States'  measure. 

In  measuring  cisterns,  reservoirs,  vats,  etc.,  the  barrel 
is  estimated  at  31i  gallons,  and  the  hogshead  63  gallons. 

A  gallon  of  water  weighs  nearly  8  J  pounds,  avoirdupois. 

A.  pint  is  generally  estimated  as  a  pound. 
22 


254      ORTON  &  Sadler's  calculator. 

LINEAK  OK  LONG   MEASURE. 
For  measuring  lengths,  distances  and  dimensions  of  objects. 


TABLE. 

12   inches 1  foot. 

3    feet lyard. 

6^  yards 1  rod. 

m  feet Irod. 

1 320    rods 1  mile. 


■ 

^H 

■ 

■ 

■ 

H| 

Note.— The 

inch   is  divided 

1 

■ 

1 

I 

1 

HI 

lntoH,K,^^- 

EQUIVALENTS. 

mt. 

fur 

rd. 

yd. 

ft. 

m. 

1    : 

=  8 

= 

320 

= 

1760 

=  6280 

= 

63360 

1 

= 

40 

= 

220 

=     660 

r= 

7920 

1 

^ 

5i 

1 

=       16^ 
=         3 
1 

= 

198 
36 
12 

Scale  of  units :— 12,  3,  5i,  40,  8. 

Note. — Cloth  Meamre  is  practically  out  of  use.  In  measuring  goods 
sold  by  the  yard,  the  yard  is  divided  into  halves,  fourth^^  eighths,  and 
sixteenths.  At  United  States  Custom  Houses,  in  estimating  duties,  the 
yard  is  divided  into  tenths  and  hundredths. 

^  FOR  MEASURING  HEIGHTS  AND  DISTANCES. 

3  inches 1  palm,     l     9  inches 1  span. 

4  "      Ihand.     |    3t'o  feet 1  pace. 

MARINER'S   MEASURE. 
Table  used  by  mariners  in  calculating  distances  on 
water,  and  the  speed  of  vessels. 


9  in 1  span. 

8  spans,  or  6  ft....l  fathom. 
120  fath 1  cable's  length. 


7  J  cables 1  mile  or  knot. 

51  ft.  nearly....  1     "    "      " 
3  miles 1  league. 


Note.— The  number  of  knots  of  the  log  line  run  off  in  half  a  minute 
indicates  the  number  of  knots  of  distance  a  vessel  goes  per  hour. 


WEIGHTS  AND  MEASURES.  255 

SURVEYOR'S  LONG  MEASURE, 
For  measuring  boundaries  of  land,  areas,  railroads,  canals. 

Tjort  Inches I  link.  I    4  Rods 1  chain. 

25  Links 1  rod.    |  SO  Chains 1  mile. 

EQUIVALENTS. 
mi,         ch.  rd,  I.  hi, 

1    =    80    =     320    =    8000    ^    63360 
1     ^        4    =r      100    -=        792 
1     r=        25    --=        198 

1    =  7.92 

Scale  of  Units  :— 7.92,  25, 4,  80. 

10  chains  long  by  1  broad,  or  10  square  chains,  1  acre. 

Gunter's  Chain,  which  is  the  unit  of  measure  used 
by  surveyors,  is  QQ  feet  long,  consisting  of  100  links. 

Measurements  are  recorded  in  chains  and  hundredths. 
Latterly  a  steel  measuring  tape  100  feet  long,  with  each 
foot  divided  into  tenths,  is  used  by  engineers  as  a  sub- 
stitute for  the  cumbersome  chain. 

Note. — By  scientific  persons  and  revenue  officers,  the  inch  is  divided 
into  tenths^  hundredlhs,  etc.  Anionji;  mechanics,  the  inch  is  divideJ 
into  nffhths.  The  division  of  the  inch  into  12  parts,  callel  lines,  is 
not  now  in  use 

A  standard  Kiiulisb  mile,  wliich  is  the  measure  that  we  use,  is  5280 
feet  in  U-U!;ih,  ITiJO  yards,  or  320  rods.  A  strip,  one  rod  wide  and  ono 
mile  long,  is  tN\o  acr(-s.  By  this  it  is  easy  to  calcnlate  the  quantity 
of  laud  taken  up  by  roads,  and  also  how  much  is  wabteJ  by  fenced. 

TABLE 

For  Geographical  and  Astronomical  Calculations. 

1  Geographic  mile 1 .15  statute  miles. 

3  *'  "    1  league. 

60            "            "   or  69.16  statute  miles.l  degree. 
360  Degrees Circumference  of  the  earth. 

Note. — The  earth's  circumference  is  24,8553/^  miles,  nearly.  The 
nautical  mile  ijG'J75|  feet,  or  71)5^  fiet  longer  than  the  common  mile. 


256      orton  &  Sadler's  calculator. 


SUKFACE   OB 
SQUABE  MEASUKE 

Used  in  ascertaining  the 
extent  of  surfaces,  such 
as  land,  boards,  plaster- 
ing, paving,  etc. 


TABLE. 

144    Square  Inches  (sq.  in.). ..I  square  foot,  sq.  ft. 

9    Square  Feet 1  square  ysird,sq.yd. 

30i  Square  Yards 1  sq.  rod  or  perch,  sq.  rd.;  P. 

160    Square  Rods 1  acre,  A. 

640    Acres 1  square  mile,  sq.  mi. 

»q.mi.  A.        sq.  rd.        sq.  yd.  sq.ft.  sq.  in. 

1  —  640  =:  102400  =  3097600    =  27878400  ==4014489600 

1  =^       160  =:^       4840   =       43560  =      6272640 

1=  30i=  2721-==  39204 

1    ~  9  ==  1296 

1  ^  144 

Scale  of  Units  :— 144,  9,  30i,  40,  4,  640. 

Measure  209  feet  on  each  side,  and  you  have  a  square 
acre  within  an  inch. 

Note. — The  following  gives  the  comparative  size,  in  square  yards, 
of  acres  in  different  countries: 

English  acre,  4840  square  yards;  Scotch,  G150;  Irish,  7840;  Ham- 
bnrgh,  11,545;  Amsterdam,  9722;  Dautzic,  G650;  Frunce  (hectare), 
11,900;  Prussia  (morgen),  3053. 

This  difference  hhould  be  borne  in  mind  in  reading  of  the  products 
per  acre  in  different  countries.     Our  laud  measure  is  that  of  England. 

Artificers  estimate  their  work  as  follows: 

By  the  square  foot ;  as  in  glazini,',  stone-cutting,  etc. 

By  the  square  yard,  or  by  the  square  of  100  square  feet; 
as  in  plastering,  flooring,  roofing,  paving,  etc. 

One  thousand  shingles,  averaging  4  inches  wide,  and 
laid  5  inches  to  the  weather,  are  estimated  to  be  a  square. 


WEIGHTS  AND   MEASURED. 


257 


SURVEYORS'    SQUARE    MEASURE, 
Used  for  measuring  the  area  or  contents  of  fields,  farms, 
and  government  lands. 

TABLE. 

625  Square  Links  (s<7.  ^.) 1  pole,  P. 

16  Poles I  square  chain,  6<7.  ch, 

10  Square  Chains 1  acre,  A. 

640  Acres 1  square  mile,  sq^mi. 

36  Square  Miles  (6  miles  sq.)....l  township,  Tp, 

EQUIVALENTS. 
Tp,    sq.  mi.       A.  sq.  ch.  P.  sq.  I. 

1  ==  36  ==  23040  =  230400  ^  3686400  ==  2304000000 

1  =   640  =   6400  =  102400  ^   64000000 

1  =    10  ■=    160  ^     10000 

11=      1(3  z=:       1000 

I  =  625 

Scale  of  Units  :— 625, 16, 10, 640,  36. 

The  acre  is  the  unit  of  land  measure. 

GOVERNMENT  LAND   MEASURE. 

A  township — 36  sections,  each  a  mile  square. 

A  section— 640  acres. 

A  quarter  section,  half  a  mile  square — 160  acres. 

An  eighth  section,  half  a  mile  long,  north  and  south, 
and  a  quarter  of  a  mile  wide — 80  acres. 

X  sixteenth  section,  a  quarter  of  a  mile  square— 40  acres. 

The  sections  are  all  numbered  1  to  36,  commencing  at 
the  northeast  corner,  thus : 

The  sections  are  all  di- 
vided into  quarters,  which 
are  named  by  the  cardinal 
points,  as  in  section  1 .  The 
quarters  are  divided  in  the 
same  way.  The  descrip- 
tion of  a  forty-acre  lot 
would  read:  The  south 
half  of  the  west  half  of  the 
south-west  (quarter  of  sec- 
tion 1  in  townsliip  24,  north 
of  range  7  west,  or  as  the  case  might  be;  and  sometimes 
will  fall  short,  and  sometimes  overrun  the  number  of 
acres  it  is  supposed  to  contain. 


6 

6 

4 

3 

2 

NW  1  NE 
SW  1  SE 

7 

8 

9 

10 

11 

12 

18 

17 

16* 

15. 

14 

13 

19 
30 

20 

29 
32 

21 

28 
33 

22 
27 
34 

23 
35 

24 
25 

131 

36 

258 


ORTOIS^  «  SADLER  8    CALCULATOR. 


CONTENTS   OF    FIELDS   AND   LOTS. 

For  the  convenience  of  farmers  and  others  who  desire 
lay  off  small  lots  of   land   for  sale,   or  to  ascertain  the 
amount  of  land  in  fields,  the  following  table  is  prepared, 
and  will  be  found  accurate  : 


52^  ft.  square  or  2722^  square  ft.  - 


=  1  A 


735  -        "     " 

5445 

10890 

120-.V  *'        "     " 

14520 

U7^^  a           "       <^ 

21780 

2085  *        "     " 

43560 

10  rods  X    16  rods 

z=:   1 

8     "     X    20     " 

==::    1 

6     "     X    32     " 

=    1 

4    "     X    40     " 

zr=    1 

5  yds.  X  968  yds. 
10    ^'    X  484    " 

1 

20    "     X  242    " 

=    1 

40     "     X  121     *' 

=    1 

80    "     X    60.^  " 

=    1 

70    "     X    69;^  " 

=    1 

220  feet  X  198  feet 

=    1 

440    "     X    99    " 

=    1 

^ig  of  an  acre. 


1      ^l      a      n 

=  i      *'      "      u 
_  ^      u      a      a 

^   1 

110  feet  X  396  feet  =  1  A 


60     "     X  726     " 

z=    1 

120     "     X  363     " 

=  1 

240     "     X  18U  " 

z=^    1 

200    "     X  108^9/' 

=  i 

100     "     X145,V' 

z=    J 

100     "     X  108^0-" 

—  i 

25  ft.  X  100  ft.  = 

.0574 

25    "    X   110    "    := 

.0631 

25   "  X  120   "   = 

.0688 

25   "  X  125   "   = 

.0717 

25  '*  X  150  "   = 

.109 

TABLE 
Showing  the  number  of  Rails,  Stakes  and  Posts  required 
for  10  rods  of  Post-and-Rail  Fence. 


Length 

of  rail. 

ft. 

Length 

of  pan  el. 

ft. 

No.  of 
panels. 

No.  of 
posts. 

Number  of  rails. 

6  rails  high.  I  7  rails  high. 

10 
12 
14 

163^ 

8 
10 
12 
14^ 

20% 

21 
17 
14 
12 

124 
99 

83 
68 

145 
116 
96 

79 

Note. — In  arranging  the  above  table  12  inches  lap  have  been  al- 
lowed. The  greater  the  lap,  the  stronger  and  more  durable  the  fence. 
To  ascertain  the  number  of  rails,  etc.,  for  any  desired  length  offence, 
multiply  the  numbers  given  in  the  above  table  by  the  length  in  feet, 
and  point  off  one  figure  from  the  left,  and  you  have  the  desired  result. 


WEIGHTS   AND   MEASURES 


259 


TKOY  WEIGHT 

Is  used  for  weighing  gold, 
silver,  platina,  and  pre- 
cious stones,  except  dia- 
monds ;  also  in  philosoph- 
ical experiments. 

TABLE. 
24  grains  {gr.)  make  1  pen- 
nyweight, pwt. 
20  pwt.  make  1  ounce,  oz. 
12  oz.        "      1  i)Ound,  Ih. 


^'t-iiffiS^gntii 


EQUIVALENTS. 

Ih. 

OZ.                      pivt. 

1 

=        12        =        240 

1         =          20 

1 

gr. 
6760 
480 
24 
Scale  of  units:— 24,  20,  12. 

Note — Troy  "Weight  contains  5760  grains  to  the  pound,  or  1240 
grains  less  tlian  the  avoirduiwis  pound.  In  mixing  medicines,  not 
Uquid,  apotliecaries  use  Troy  grains. 

DIAMOND    WEIGHT    TABLE. 

16  parts 1  grain. 

4  gr 1  carat. 

1  K 3i  Troy  grains,  nearly  Zh 

The  word  carat  is  used  to  express  the  fineness  of  gold, 
and  means  ^^  part.  Pure  gold  is  said  to  be  24  carats  fine ; 
if  there  be  22  parts  of  pure  gold  and  2  parts  of  alloy,  it  is 
said  to  be  22  carats  fine.  The  standard  of  American  coin 
is  nine-tenths  pure  gold,  and  is  worth  $20.67.  What  is 
called  the  neiv  standard,  used  for  watch  cases,  etc.,  is  18 
carats  fine.  The  term  carat  is  also  applied  to  a  weight  of  3i 
grains  Troy,  used  in  weighing  diamonds;  it  is  divided  into 
4  parts,  called  grains ;  4  grains  Troy  are  thus  equal  to  5 
grains  diamond  weight. 


260 


ORTON    &   SADLER  S   CALCULATOR. 


PAPER  AND  BOOKS. 


The  followiDg  de- 
nominations of  meas- 
ure are  used  by  the 
paper  manufacturer, 
book  and  stationery 
trade. 

TABLE. 

24  Sheets I  quire. 

20  Quires 1  ream. 

2  Reams 1  bundle. 

5  Bundles 1  bale. 

1  Bale  contains  200  quires 
or  4800  sheets. 


SIZES   OF  PAPER. 

Paper  manufactured  to  order  can  be  made 
any  desired  size.  The  following  are  regular  or 
trade  sizes,  and  can  generally  be  found  in  stock 
at  any  of  the  wholesale  paper  houses. 


WRITING   PAPERS— FLAT   CAP. 


Name.  Size,  In. 

Law  Blank 13x16 

Flat  Cap 14X17 

Crown 15X19 

Demy 16X21 

Folio  Post 17X22 

Check  Folio 17X24 

Double  Cap 17X28 

Ex.  Size  Folio....  19X23 


Name.  Size,  In. 

Medium 18  X23 

Royal 19  X24 

Super  Royal... 20  X28 

Imperial 22  X30 

Elephant 22^X271 

Columbia 23  X33j 

Atlas 26  X33 

Doub.  Elephant.  26  X40 


PRINTERS   AND  STATIONERS.  261 

WRITING  PAPERS— FOLDED, 
j^ame.  Size,  In.  |  Name.  Size,  In. 


Billet  Note 6   X  8 

Octavo  Note 7   X   9 

Commercial  Note  .8   X 10 


Bath  Note 8^x14 


Letter 10   X16 

Oommerc'l  Let.ll   x  11 
Packet  Post....  11^X18 


Packet  Note 9   Xll    Ex.  Pack.  Post. lUxlSJ 


Foolscap 12^X16 


PRINTING    PAPER. 

Used  in  Printing  Newspapers  and  Books. 
Name.  Size,  In.   ]  Name.  Size,  In. 

Medium 19x 24  i  Double  Medium. .  24x  38 

Royal 20x25  !  Double  Royal. . . ,  26x40 

Super  Royal 22x28!  Doub.  Sup.  Royal .  28  X  42 

Imperial 22x32        "         "  "     .29x43 

Medium-and-half.. 24x30    Broad  Twelves. .  .23x41 
Small  Doub.  Med. 24x36  ■  Double  Imperial.  .32x46 

COPYING. 

In  the  copying  of  papers,  manuscripts,  and 
documents  for  official  record,  clerks  and  copyists 
are  usually  paid  by  the  folio. 

A  folio  varies  in  quantity  in  different  States 
and  sections  of  the  world,  but  is  generally  esti- 
mated from  75  to  100  words. 

PRINTING— TYPE-SETTING. 
Printers  generally  charge  for  setting  type,  or 
the  composition  of  matter,  as  it  is  technically 
termed,  by  the  number  of  ems  it  contains,  rated 
by  the  1000  enis;  an  em  is  the  square  of  the 
body  of  the  type. 


262        ORTON   &  SADLER^S    CALCULATOR. 

PRINTING-PRESS-WORK. 

Press-work  is  generally  charged  by  the  token 
of  250  impressions,  or  125  sheets,  printed  on 
both  sides.  The  value  or  cost  of  press- work 
depends  upon  the  style,  quantity,  and  quality  of 
ink  used. 

MISCELLANEOUS  TABLES. 

FOR     COUNTING     CERTAIN     ARTICLES. 

12  Units  or  pieces 1  dozen. 

20      '*  "      1  score. 

12  Dozen 1  gross. 

12  Gross .1  great-gross. 

BOOKS. 

Names  and  Sizes  as  classified  hy  Publishers, 
The  number  of  folds  and  pages  in  a  single 
sheet  when  manufactured. 

Name  of  book.  Ji^illtol'S.Vi.  Contain. 

Folio 2  leaves,...   4  pages. 

Quarto  or  4to 4  "      8  '* 

Octavo  or  8vo 8  ''      16  '' 

Duodecimo  or  12  mo 12  **  ....  24  " 

16  mo 32  "  ....64  " 

18    '*  * 18  "  ....36  *' 

24    "   24  "  ....48  " 

32    "   32  "  ....64  '' 

*  Note. — This  book  is  an  18mo.,  there  being  36  pages 
to  the  sheet.  The  terms  folio,  quarto,  octavo,  etc.,  denote 
the  number  of  leaves  in  which  a  sheet  of  paper  is  folded. 

The  marks  A,  B,  C ;  1,  2,  3 ;  lA,  2A ;  l-%  2*,  etc.,  oo- 
casionally  found  at  the  bottom  of  pages,  are  what  printers 
term  signature  marks,  thus,  3*,  being  printed  for  the  direc- 
tion of  binders  in  folding  the  sheets. 


PRINTERS   AND   STATIONERS.  263 

Printers  and  stationers  generally  procure  their 
supplies  of  paper  by  the  quantity,  the  cost  per 
ream  varying  according  to  the  quality  and 
weight. 

The  table  on  page  264  will  be  found  invaluable 
in  making  up  their  estimate  for  small  job  work, 
as  it  shows  at  a  glance  the  cost  per  quire  of  paper 
purchased  at  15  to  30  cents  per  pound,  and 
weighing  from  10  to  60  pounds  to  the  ream. 

Example. — What  does  one  quire  of  paper 
cost  purchased  at  23  cents  per  pound,  and  weigh- 
ing 40  pounds  to  the  ream  ?    Am,  46  cents. 

Explanation. — Refer  to  the  weight  column 
on  the  left  and  the  purchasing  price  at  the  top, 
and  you  have  the  cost  per  quire  shown  in  the 
purchasing  column  on  the  line  with  the  weight. 

SHOEMAKER'S   MEASUBE. 
3  Barleycorns  or  sizes 1  inch. 

Number  one,  children's  measure^  is  4|  Inches,  and  that 
every  additional  number  calls  for  an  increase  of  i  of  an 
inch  in  length.  Number  one,  adults*  measure,  is  8^ 
inches  long,  with  a  gradual  increase  of  \  of  an  inch  for 
additional  numbers,  so  that,  for  example,  number  ten 
measures  Hi  inches.  This  measure  corresponds  to  the 
number  of  the  last^  and  not  to  the  length  of  the  sole. 


TABLE 
To  Ascertain  the  Cost  of  One  Quire  of  Paper. 


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264 


MEASUREMENT  OF  TIME. 

TIMB  IS  THE  MEASURB  OP  DURATION 


We  have  in  this  eugraving  a  represeniatiou  of 
the  magnificent  transit  instrument  used  in  the 
Paris  Observatory.  It  is  made  on  the  same  model 
as  the  celebrated  one  in  the  Observatory  at  Green- 
wich. These,  and  the  one  at  the  National  Observa* 
tory  at  Washington,  are  the  finest  in  the  world. 
The  instrument  is  used  for  the  purpose  of  determin- 
ing the  instant  of  time  a  heavenly  body  passes,  Of 
qualDos  a  tr^vndt  across  the  meridian. 
23  265 


266      ORTON  «fe  Sadler's  calculator. 
MEASUREMENT    OP    TIME. 

TABLE. 

60  seconds 1  minut*. 

60  minutes 1  hour. 

24  hours 1  day. 

7  days 1  week. 

28  days 1  lunar  month. 

28,  29.  30,  or  31  days 1  cal.  mouth. 

12  cal.  months  1  year. 

365  days* 1  com,  year. 

366  days* 1  leap  year,* 

365W  days 1  Julian  year. 

365  d,,  6  h.,  48  m,,  49  8 1  solar  year. 

365  d.,  6  h.,  9  m,,  12  s 1  siderial  year. 

10  years 1  decade. 

10  decades,  or  100  years 1  century. 

EQUIVALENTS. 
yr.  da.  hr.  min.  mc. 

1    =    365V^    ==     8766    =     625960     =    31557600 
1        =        24    =        1440     =  86400 

1     =  60    =  3600 

1     =  60 

Scale  of  units :— 60,  60,  24,  365J4. 

»  NoTK.— The  common  year  thus  consists  of  365  days.  Once  in  4  years,  however, 
one  day  Is  added  to  February,  making  366  days ;  and  thu3,  each  year  averages 
365K  days.  The  longest  year  is  called  Bissextile,  or  Leap  year.  Centuries  divisi- 
ble by  400,  and  other  years  divisible  by  4,  are  leap  years, 

Tn  business  transactions  30  daj'S  are  considered  1  month.  The  civil 
day  begins  and  ends  at  12  o'clock,  midnight. 

In  dating  events,  astronomers  calculate  the  day  as  beginning  and 
ending  at  12  o'clock,  noon. 

CIRCUIiAK    MEASURE, 
Or,  Divisions  of  the  Circle, 
Used  in  Astronomy,  Geography,  Navigation,  and  Surveying;  also  for 
calculating  the  differences  of  time. 

TABLE. 

60  seconds  (") 1  minute ' 

60' 1  degree ° 

30° 1  sign S. 

12  S.,  or  360° 1  circle C* 


a 

S.             o        ' 

tf 

1 

=  12  =  360  =  21600 

=  1296000 

1  =   30  =   1800 

=   108000 

1  =s    60 

=    3600 

1 

=      60 

Scaleof  units  :■ 

-60,  60,  30, 12. 

*A  senii-eircumferenoe  is  M  of  a  circle,  180° 
A  quadrant  *'  J4  "  "      "       ^^ 

A  sextant  "><"••      "       &P 


MEASUREMENT   OF    TIME.  267 

The  greatest  distance  across  a  circle  is  called  its  diameter.  The  dis- 
tance around  it  is  called  its  circumference.  Any  part  of  the  circum- 
ference is  called  an  arc. 

LONGITUDE   AND    TIME. 

TABLE. 

For  a  difiference  of  |  There  is  a  difference  of 

l^iu  Long 4  m.  in  Time. 

1'       "        4  sec.      " 

V      «        ^mc.  " 

1  hr.  in  Time 15°  in  Long. 

Im.        «       15' 

Isec.      "       lb" 

NoTB. — Add  difference  of  time  for  places  east  and  subtract  it  for  places  weM  of  any 
given  place. 

HOW   TO    ASCERTAIN 

The  Difference  of  Time  between  Cities. 

BASIS    OF   CALCULATION. 

360  degrees  =  1  revolution  of  the  earth,  or  1  day. 
1440  minutes  =  1        "  "     "        ''      "  1    " 

1440  -7-  360  =  4  minutes,  or  1  degree. 

Refer  to  your  map  and  notice  the  difference  in  degrees  of  longitude 
between  places.  Multiply  the  number  of  degrees  by  4 ;  the  product 
will  be  the  difference  in  time. 

Degrees  of  Longitude  East,  time  increases. 
*'        "  "        West,    "    decreases. 

PROBLEM. 

When  it  is  12  o'clock  noon  at  Washington,  what  time  Is  it  at 
Boston?  A  ns.,  12.24. 

Per  map,  difference  in  degrees,  6  east,  which  increase  the  time 
6  X  4  =  24  differences  in  time,  or  24  minutes  past  12. 

PROBLEM. 

When  it  is  12  o'clock  noon  at  Washington,  what  is  the  time  in  San 
Frjincisco,  Califonii  i  ?  Ans.,  8.58  a.  m. 

Per  map,  difference  45^  degrees,  West;  Ab}/^  X  4  =  182  min.,  or 
3h.  2m.  less.  Ans.,  8  o'clock,  58  min.,  a.  m. 

A  telegram  sent  from  Washington  to  San  Francisco  at  12  m.  will  be 
received  at  9  o'clock  a.  m.  (i.  e.,  three  hours  lifore  it  is  stnt)^  calculat- 
ing Son  i"  rancisco  time. 


268      ORTON  &  Sadler's  calculator. 
TABLE 

For  ascertaining  the  number  of  days  between  two  dates. 


Jan. 

Feb. 

Mar. 

Apl. 

May 

June 

July 

Aug. 

Sept. 

Oct. 

Nov. 

Dec. 

1 

32 

60 

91 

121 

152 

182 

213 

244 

274 

305 

335 

2 

33 

61 

92 

122 

153 

183 

214 

245 

275 

306 

3:56 

3 

34 

62 

93 

123 

154 

184 

215 

246 

276 

307 

337 

4 

35 

63 

94 

124 

155 

185 

216 

247 

277 

30S 

338 

5 

30 

64 

95 

125 

156 

186 

217 

248 

278 

309 

339 

6 

37 

65 

96 

126 

157 

187 

218 

249 

279 

310 

340 

7 

38 

66 

97 

127 

158 

188 

219 

250 

280 

311 

341 

8 

39 

67 

98 

128 

159 

189 

220 

251 

281 

312 

342 

9 

40 

08 

99 

129 

160 

190 

221 

252 

282 

313 

343 

10 

41 

09 

100 

130 

101 

191 

222 

253 

283 

314 

344 

U 

42 

70 

101 

131 

162 

192 

223 

254 

284 

315 

345 

12 

43 

71 

102 

132 

163 

193 

224 

255 

285 

316 

346 

Vi 

44 

72 

103 

133 

104 

194 

225 

256 

286 

317 

347 

14 

45 

73 

104 

134 

165 

195 

226 

257 

287 

318 

348 

15 

46 

74 

105 

135 

166 

196 

227 

258 

288 

319 

349 

16 

47 

75 

lOo 

136 

167 

197 

228 

259 

289 

320 

350 

17 

48 

76 

107 

137 

168 

198 

229 

260 

290 

321 

351 

18 

49 

77 

108 

138 

169 

199 

230 

261 

291 

322 

352 

VJ 

50 

78 

109 

139 

170 

200 

231 

262 

292 

323 

353 

20 

51 

79 

110 

140 

171 

201 

232 

263 

293 

324 

354 

21 

62 

80 

111 

141 

172 

202 

233 

264 

294 

325 

355 

22 

53 

81 

112 

142 

173 

203 

234 

265 

295 

326 

356 

23 

54 

82 

113 

143 

174 

204 

235 

266 

296 

327 

357 

24 

55 

83 

114 

144 

175 

205 

236 

267 

297 

328 

358 

25 

56 

84 

115 

145 

176 

206 

237 

268 

298 

329 

359 

26 

57 

85 

116 

146 

177 

207 

238 

269 

299 

330 

300 

27 

58 

86 

117 

147 

178 

208 

239 

270 

300 

331 

361 

28 

69 

87 

118 

148 

179 

209 

240 

271 

301 

332 

362 

29 

88 

119 

149 

180 

210 

241 

272 

302 

333 

363 

80 

89 

120 

150 

181 

211 

242 

273 

303 

334 

364 

31 

90 

151 

212 

243 

304 

365 

Note. — To  find  from  the  above  table  the  number  of  days  between 
two  dates,  we  give  the  following — 

RuLR  I. —  WJien  the  dates  are  in  the  same  year,  subtract  the  number 
of  days  of  the  parlier  date  from  the  number  of  days  of  the  later  date; 
the  result  will  be  the  number  of  days  required. 

II.  When  the  dates  are  in  consecutive  years,  subtract  the  number  of 
days  of  the  earlier  date  from  365,  and  add  to  the  remainder  the  number 
of  days  of  the  later  date;  the  residt  will  be  the  number  of  days  required. 

When  the  year  is  a  leap  year,  add  one  day  to  the  result. 


MEASUREMENT   OF    TIME. 


269 


TABLE 

Showing  the  number  of  days  from  any  day  in  one  month 
to  the  same  day  in  any  other. 


4 

03 

t 

< 

eg 

p 
s 

>, 

s 

be 
< 

^ 

A 
^ 

1 

> 
o 
55 

365 

31 

59 

90 

120 

151 

181 

212 

243 

273 

304 

334 

365 

28 

59 

89 

120 

150 

181 

212 

242 

273 

306 

337 

365 

31 

61 

92 

122 

153 

184 

214 

245 

275 

306  334 

365 

30 

61 

91 

122 

153 

183 

214 

245 

276  304 

335 

365 

31 

61 

92 

123 

153 

184 

214 

245  273 

304 

334 

365 

30 

61 

92 

122 

153 

184 

2151243 

274 

304 

335 

365 

31 

62 

92 

123 

153 

184  212 

243 

273 

304 

334 

365 

31 

61 

92 

122 

153|181 

212 

242 

273 

303 

334 

365 

30 

61 

92 

123 1 151 

182 

212 

243 

273 

304 

335 

365 

31 

61 

921120 

151 

181 

212 

242 

273 

304 

334 

365 

31 

62 

90 

121 

151 

182 

212 

243 

274 

304 

335 

Jan.. . 
Feb... 
March 
April. 
May . . 
June . 
July  . 
Aug... 
Sept . . 
Oct . . . 
Nov.., 
Dec... 


334 
303 
275 
244 
214 
183 
153 
122 
91 
61 
30 
365 


Note. — Find  in  the  left-hand  column  the  month  from  any  day  of 
which  you  wish  to  compute  the  number  of  days  to  the  same  day  in 
any  other  month  ;  then  follow  the  line  along  until  under  the  desired 
month,  and  you  have  the  required  number  of  days. 

Example. — How  many  days  from  March  15  to  July  15  ? 

Ans.,  122  days. 

In  leap-year,  when  the  month  of  February  occurs  in  the 
calculation,  one  day  extra  must  be  added. 

Example. — 1876.  How  many  days  from  January  13 
to  May  13  ?  Ans.,  Per  table,  120 ;  one  day  added  for  leap- 
year  =  121  days. 


ASTRONOMICAL  CALCULATiaNS 


A  scientific  method  of  telling  immediately  what  dau 
of  the  week  any  date  transpired  or  will  transpire^ 
from  the  commencement  of  the  Christian  Era,  for 
the  term  of  three  tnousand  years. 

MONTHLY   TABLE. 

The  ratio  to  add  for  each  month  will  be  found 
in  the  following  table: 


Ratio  of  October  is 3 

Ratio  of  May  is 4 

Ratio  of  August  is ^.6 

Ratio  of  March  is 6 

Ratio  of  February  is 6 

Ratio  of  November  is 6 


Ratio  o€  June  is 0 

Ratio  of  September  is 1 

Ratio  of  December  is 1 

Ratio  of  April  is 2 

Ratio  of  July  is 2 

Ratio  of  January  is ..3 

Note. — On  Leap  Year  the  ratio  of  January  is  2,  and 
the  ratio  of  February  is  6.  The  ratio  of  the  other  ten 
months  do  not  change  on  Leap  Years. 

CENTENNIAL   TABLE. 

The  ratio  to  add  for  each  century  will  be  found 
In  the  following  table: 
o    200,    900,  1800,  2200,  2600,  3000,  ratio  is 0 


800,  1000, 


ratio  is 6 


5-    400,  1100,  1900,  2300,  2700,  ratio  is 6 

^    600   1200,  1600,  2000,  2400,  2800,  ratio  is 4 

2     600    1300,       ratio  is 3 

000,  700,  1400,  1700,  2100,  2500,  2900,  ratio  is 2 

loo,  800,  150a  ratio  IB 1 

270 


ASTRONOMICAL  CALCULATIONS.      271 

Note. — The  figure  opposite  each  century  is  itg  ratio; 
thus  the  ratio  for  200,  900,  etc.,  is  0.  To  find  tho  ratie 
of  any  century,  first  find  the  century  in  the  above  table, 
then  run  the  eye  along  the  line  until  you  arrive  at  the 
end;  the  small  figure  at  the  end  is  its  ratio. 

METHOT)   OP   OPERATION. 

Rule.* — To  the  given  year  add  its  fourth  part. 
Tweeting  the  fractions  ;  to  this  sum  add  the  day  oj 
the  month;  then  add  the  ratio  of  the  month  and 
the  ratio  of  the  century.  Divide  this  sum  hy  7 ;  the 
remfiainder  is  the  day  of  the  week^  counting  Sunday 
a»  the  first^  Monday  as  the  second^  Tuesday  as  the 
third,  Wednesday  as  the  fourth,  Thursday  as  the 
fifth,  Friday  as  the  sixth,  Saturday  as  the  seventh; 
the  remainder  for  Saturday  will  he  0  or  zero. 

Example  1. — Required  the  day  of  the  week 
for  the  4th  of  July,  1810. 

To  the  given  year,  which  is 10 

Add  its  fourth  part,  rejecting  fractions 2 

Now  add  the  day  of  the  month,  which  is 4 

Now  add  the  ratio  of  July,  which  is 2 

Now  add  the  ratio  of  1800,  which  is 0 

DiTide  the  whole  sum  by  7.  7  |  18 — 4 

2 
We  have  4  for  a  remainder,  which  signifies  the 
fourth  day  of  the  week,  or  Wednesday. 

•  Wheti  dividing  the  year  by  4,  always  leave  off  ttie  coatories*  Wb 
#lYid«  by  i  to  find  the  number  of  Leap  Years. 


272      ORTON  &  Sadler's  calculator. 

Note. — In  finding  the  day  of  the  week  for  the  present 
century,  no  attention  need  be  paid  u)  the  centennial  raUo^ 
as  it  is  0. 

Example  2. — Required  the  day  of  the  week 
for  the  2d  of  June,  1805. 

To  the  given  year,  which  ia 6 

Add  its  fourth  part,  rejecting  fractions ..,..  1 

Now  add  the  day  of  the  month,  which  is 2 

Now  add  the  ratio  of  June,  which  is 0 

Divide  the  whole  sum  by  7.  7  |  8^i 

T 

We  have  1  for  a  remainder,  which  signifies  the 
first  day  of  the  week,  or  Sunday. 

The  Declaration  of  American  Independence 
was  signed  July  4,  1776.  Required  the  day  of 
the  week. 

To  the  given  year,  which  is 76 

Add  its  fourth  part,  rejecting  fractions 19 

Now  add  the  day  of  the  month,  which  is 4 

Now  add  the  ratio  of  July,  which  is 2 

Xow  add  the  ratio  of  1700,  which  is 2 

Divide  the  whole  sum  by  7.  7  |  103 — 6 

14 

We  have  5  for  a  remainder,  which  signifies  the 
fifth  day  of  the  week,  or  Thursday. 

The  Pilgrim  Fathers  landed  on  Plymouth  Rock 
Doc  20,  ltf20.     Reqmired  the  day  of  the  w^eeV 


ASTRONOMICAL   CALCULATIONS.  273 

i'o  the  given  year,  which  is JiO 

Add  its  fourth  part,  rejectiug  fractions 6 

How  add  the  day  of  the  monih,  which  is 20 

Now  add  the  ratio  of  December,  which  is 1 

Now  add  the  ratio  of  1600,  which  is 4 

Divide  the  whole  sum  by  7.  7  |  50- -1 

7 

We  have  1  for  a  remainder,  which  signifies  th* 
first  day  of  the  week,  or  Sunday. 

On  what  day  will  happen  the  8th  of  January, 
1815?     Ans.  Sunday. 

On  what  day  will  happen  the  4th  of  May,  1810? 

On  what  day  will  happen  the  3d  of  December, 
1423?     Ans,  Friday. 

On  what  day  of  the  week  were  you  born? 

The  earth  revolves  round  the  sun  once  in  365 
days,  5  hours,  48  minutes,  48  seconds ;  this  period 
is,  therefore,  a  Solar  year.  In  order  to  keep  pace 
with  the  6o\sLT.  year,  in  our  reckoning,  we  make 
©very  fourth  to  contain  366  days,  and  call  it  Leap 
Year.  Still  greater  accuracy  requires,  howevei, 
that  the  leap  day  be  dispensed  with  three  times 
(D  every  400  years.  Hence,  every  year  (except 
the  centennial  years)  that  is  divisible  by  4  is  a 
Leap  Year,  and  every  centennial  year  that  i? 
divisible  by  400  is  also  a  Leap  Year.  The  next 
i^«Dtenn.'al  year  that  will  be  a  Leap  Year  h  2000 


274 


ORTON   &   SADLER  S   CALCULATOR. 


MONEY  OP  THE  UNITED  STATES 

Is  the  measure  of  value 
of  all  kinds,  such  as 
property,  merchandise, 
services,  etc.  It  is  the 
medium  of  exchange  in 
business. 

Coin  or  Specie  is 
metal  stamped  and  au- 
thorized by  government 
to  be  used  as  money. 

Paper  Money  con- 
sists of  notes  issued  by 
the  United  States  Treas- 
ury and  banks,  and  used 
as  money.  United  States 
money  is  the  legal  currency  of  the  United  States. 

U.    S.    MONEY. 

10  mills  {M,) 1  cent ct.  or  ^. 

10^ 1  dime 

10  dimes ...1  dollar i>.or$. 

10  dollars 1  eagle E. 

Note. — The  mill  is  not  coined. 
The  Coin  of  the  United  States  consists  of  gold,  silver, 
nickel  and  bronze,  and  as  fixed  by  the  "  New  Coinage  Act" 
of  1873  is  as  follows : 

Gold.  The  double-eagle,  eagle,  half-eagle,  quarter- 
eagle,  three-dollar,  and  one-dollar  pieces. 

Silver.    The  jTmcZ^-doUar,  half-dollar,  quarter-dollar, 
the  twenty-cent,  and  the  ten-cent  pieces. 
Nickel.    The  five-cent  and  three-cent  pieces. 
Bronze.    The  one-cent  piece. 

Note. — The  term  shilling  is  frequently  used  in  the  United  States  in 
stating  the  price  of  articles,  and  it  indicates  old  divisions  or  equivalents 
of  parts  of  the  dollar.  Its  value  vnries  in  dilferent  States  as  follows: 
In  the  New  England  States,  and  in  Indiana,  Illinois,  Missouri,  Missis- 
sippi, Texas,  Virginia,  Kentucky,  and  Tennessee,  ls.=:16%  cts.,  and  $1. 
=6s, ;  in  New  York,  Ohio,  Michigan,  and  North  Caiolina,  l6.=123/^ct8. 
and  $1  .=8s. ;  in  Pennsylvania,  New  Jersey,  Delaware,  and  Maryland, 
l8.=13i^  cts.,  and  $i.=7^8. ;  in  Georgia  and  South  Carolina,  l8.= 
21 1  cts.,  and  $1 .  =i%B.  These  rates  are  liable  to  variations  by  custom  ; 
as,  in  Illinois,  tho  shilling  is  rated  frequently  at  12}^. 

Canada  Money  consists,  like  United  States  money,  of 
dollars  and  cents.  The  Canada  coins  are  twenty-cent, 
ten-cent,  five-cent,  silver;  and  one-cent,  bronze. 


MONEY   OF   FRANCE.  275 

MONEY    OF    FRANCE. 

The  money  of  account  of  France  is  the  Franc 
of  100  Centimes  or  1000  Sous,  and  is  arranged 
on  the  decimal  system. 

The  principal  gold  coins  in  circulation  are  as 
follows :  Louis  d'or,  Forty  Franc  piece,  Twenty 
Franc  piece,  and  Six  Franc  piece. 

The  principal  silver  coins  are,  the  Crown,  } 
Crown,  i  Crown,  Five  Franc  piece,  Two  Franc 
piece.  Franc,  i  Franc  of  50  Centimes,  i  Franc 
of  26  Centimes. 

The  par  value  of  the  Franc  is  19  cents  and  3 
mills,  but  its  commercial  value  varies  according 
to  the  fluctuations  of  the  money  market.  When 
exchange  is  quoted  at  say  5.17i,  it  is  understood 
to  mean  that  5  Francs  and  17  i  Centimes  are 
equal  to  the  gold  dollar.  Tlie  gold  cost  of  any 
given  number  of  francs  is,  therefore,  ascertained 
by  dividing  that  number  by  the  quotation  or 
rate  of  exchange.  For  example,  the  cost  of  9463 
Francs  at  5.21}=9463  fr.-T-5.215  fr.=$1814.57. 
The  premium  on  gold  is  then  added  to  find  the 
cost  in  currency. 


276      ORTON  &  Sadler's  calculator. 

MONEY  OF  THE  GERMAN  EMPIRE. 

The  German  Empire  has,  within  the  past  few 
years,  issued  a  new  coin  called  the  Mark,  which 
is' now  adopted  as  the  money  of  account  of  that 
nation. 

The  gold  and  silver  coins  are  quite  numerous, 
embracing,  as  they  do,  those  in  use  in  about 
twenty-two  States,  and  are  as  follows : 

Gold, — Ducat,  Quintuple  Ducat,  Five  Thaler 
piece,  Ten  Thaler  piece.  Double  d'or,  \  Caroline, 
i  Caroline,  Caroline,  Five  Gilder  piece,  Twenty 
Mark  piece,  Ten  Mark  piece,  and  Twelve  Mark 
piece. 

Silver. — Mark,  Thaler,  Double  Thaler,  Crown 
Thaler,  J  Thaler,  Double  Gilder,  Florin,  i 
Florin,  Twelve  Grote  piece,  Grote,  Rix  Dollar, 
Crown,  Kreutzer  Groschen,  6  Pfen,  1  Schilling, 
48  Schilling  piece,  30  Kreutzer  piece,  8  Schilling 
piece. 

In  commercial  transactions  the  Mark  is  not 
generally  reckoned  at  its  par  value,  but  is  gov- 
erned by  the  quotations  which  range  at  present 
between  91  cents  and  $1.00  for  4  Marks.  The 
market  value  of  the  Mark,  in  gold,  is  found  by 
dividing  the  quotation  by  4.  For  example,  400 
Marks  at  $.95  i  =  $.955  H-  4  m.  X400  =  $95.50. 
The  premium  on  gold  is  then  added  to  find  the 
cost  in  currency.  , 


ARBITRATION  OF   EXCHANGE. 

The  method  of  finding  the  value,  in  gold  and  currency, 
of  the  moneys  of  the  principal  nations  has  already  been 
briefly  explained ;  but  the  fluctuations  of  foreign  exchange 
sometimes  render  it  to  the  advantage  of  the  merchant  to 
remit  indirectly.  For  example,  suppose  it  is  required  to 
pay  a  debt  due  in  France,  and  that  the  balance  of  trade 
between  Great  Britain  and  the  United  States  is  in  favor  of 
the  latter  nation.  This,  of  course,  would  cause  a  decline 
in  the  value  of  the  £  sterling  in  United  States  money. 
Now,  if  there  is  no  corresponding  difference  in  value  be- 
tween the  moneys  of  Great  Britain  and  France,  it  would 
be  cheaper  first  to  purchase  sterling  and  then  remit  through 
London.  Such  exchanges  are  termed  direct  and  indirect, 
as  the  case  will  indicate,  and  are  treated  under  the  rule  of 

ARBITRATION   OF  EXCHANGE. 

The  limits  of  a  work  of  this  kind  prevent  an  extended 
elucidation  of  this  subject,  and  it  is  not  thought  best  to  pre- 
sent any  set  rules  for  memorizing. 

In  the  following  examples,  with  solutions,  indirect  ex- 
change will  be  found  much  simplified.  Indeed,  when 
taken  in  the  order  of  the  different  nations  involved,  there 
will  appear  but  little  distinction  from  the  method  of  work- 
ing ordinary  exchange. 

Example  1.— When  sterling  is  quoted  at  $4.83^  U.  S.  money,  and 
26.73  fr.,  French,  how  many  franca  are  equal  to  the  gold  dollar  by  in- 
direct exchange  ? 

Solution.— U  £1=  $4,831^,  and  also  25.73  fr.,  $4.83>^  must  equal 
25.73  fr.    Therefore  25.73  fr.  -4-  $4,835  =  5.32  fr.,  ana. 

Example  2.— When  French  exchange  is  quoted  b.W/,fr,&nd  ster- 
ling 25.73  fr.,  what  is  the  value  in  gold  of  the  £  sttrling  by  indirect 
exchange  ? 

^/M^ion.— Since  $l=5.16Ufr.,  and  £1=25.73  fr.,  therefore  25.73  fr. -5- 
5.165  =  $4.98. 

Example  3.— What  would  be  the  gain  in  sterling  on  $6000  by  re- 
mitting through  France,  Germany,  and  Netherlands,  with  the  follow- 
ing quotations:  $1=  5.18  fr. ;  1.22  fr.  =  1  mark  ;  1.71  marks  =  1  guil. ; 
11.8^uil.=£l,  whcMi  the  £  sterling  by  direct  exchange  is  quoted  $4.85? 

Solution.— As  Franco  is  the  first  country,  $6000  X  6.18  fr.  =  oldSO  fr. 
-*-1.22fr.  =  24575.475  m.  -r-  1.71  m.  =14847.94  guil. -=-  11.8  guil.= 
£1202.537  =  £1262  78.  4%-|-d.,  indirect  exchange.  $G000-=-$4.85=£1237. 
1134  =  £1237  28.  3i<-f  .1.  direct  exrlmngo.  £1262  7«J.  4^d.  less  £1237 
2i.  Z]4d.  =  £25  68. 13^.  gain. 

24  277 


278     ORTON  &  Sadler's  calculator. 


J5  ?J 


.si  s^ 


;^ 


II 

^1.00 
.45,3 
.19,3 
.96,5 
.54,5 

1.00 
.91,2 
.91,8 
.91,2 
.92,5 
.26,8 
.91,8 

4.97,4 
.19,3 

4.86,6] 

3 

c 
> 

Gold  and  silver 

Gold  and  silver 

Cin\(\ 

Gold 

Gold 

Silver 

Gold 

Gold 

Silver 

Gold  and  silver 

Gold 

;  ^  2  o 

>     E>H     (^     Q 


B 

:  < 

;  -c  ^  i  -^  -s 


2  3  §  S  fc2  S  2i 
v  c  P  ^  pi!  ^  a 


VALUE  OF   FOREIGN   COINS. 


279 


«     00,  M    i 

oT  eo"  lo 

rH    CM.    O 


00^   O^  tH^  -^^ 

t-^  cm"  oo"  ro 

i    Oi     05     O    t~    ' 


eooofoooococco 
"  -*"  i-T  ■^'" 

O    05     OJ 


.2     * 


^    ^ 


is  -s  ^  s  :s 


:  ^ 
g 


SiJ 


-;         »  U         U         U 

O     eS     lU     O     O 


S    S. 


!    S  ^    «•   «    °    3 


GOLD  AND  CURRENCY. 


Gold  is  usually  represented  as  rising  and  falling,  but 
being  the  standard  of  value,  it  does  not  vary.  The  varia- 
tion is  in  the  currency  substituted  for  gold  or  specie ;  hence, 
when  gold  is  said  to  be  at  a  premium,  the  currency  or 
circulating  medium  is  made  the  standard,  while  it  is  in 
fact  below  par. 

TABLE 


Showing  the  Comparative 

Value  of 

Gold  and  Currency, 

The  Amount  in 

When  $1  in  Gold  is 

sold  for 

The  Discount  on 

Gold  \vlii 

:;h  can  be 

Currency  at 

Currency  is 

boujrht  forSlCO 

^ 

in  Cu 

'lency. 

1.01 

or       1  per  cent.  Prem. 

1.00  per  cent. 

$99.99  or    99^.1^ 

1.05 

5 

" 

" 

4.77 

« 

95.23  " 

1.10 

«      10 

«« 

" 

9.10 

" 

90.90  ' 

1.16 

"      15 

« 

«* 

13.04 

« 

86.96  » 

1.20 

«      20 

« 

" 

16.67 

u 

83.33  " 

1.25 

«      25 

u 

20.00 

tt 

80.00  ' 

80 

1.30 

«      30 

u 

23.08 

«« 

76.92  • 

75^^ 

1.331^ 

"      331^ 

" 

25.00 

'« 

75.00  " 

1.40 

"       40 

M 

28.58 

« 

71.42  ' 

71  if 

1.50 

"       50 

" 

33.33 

«* 

66.66  " 

66% 

l.GO 

"       60 

« 

37.50 

" 

62.50  " 

G2y^ 

1.66% 

"       66K 

« 

40.C0 

" 

60.00  « 

60 

1.70 

"      70 

" 

41.18 

" 

58.82  " 

58}f 

1.80 

''      80 

« 

44.45 

" 

55.55  " 

55f 

1.90 

"      90 

« 

47.37 

" 

62.63  " 

%i^ 

2.00 

"     100 

*' 

50.00 

« 

50.00  " 

2.50 

"     150 

" 

60.00 

« 

40.00  * 

40 

6.00 

"     400 

" 

" 

80.00 

" 

20.00  ' 

20 

7.50 

"     650 

« 

" 

86.67 

" 

13.33  ' 

135^ 

10.00 

"     9C0 

'« 

K 

90.00 

«♦ 

10.00 

10 

50.00 

•'  4900 

♦^ 

" 

98.00 

(( 

2.00  ' 

2 

100.00 

"  990O 

« 

" 

99.00 

" 

1.00  ' 

1 

TO  ASCERTAIN  HOW  MUCH  GOLD 

Can  be  Bought  for  a  Staled  Amount  of  Currency. 

Rule. — Add  two  ciphers  to  the  amount  of  currency  (in 
dollars),  and  divide  by  100,  increased  by  the  premium,  rate  on 
gold;  the  quotient  will  be  the  gold  sum. 

280 


BANK   ACCOUNTS. 


281 


HOW  TO  TRANSACT 

BUSINESS  WITH  BANKS. 


BANK  ACCOUNTS. 

HINTS  TO    MANY   AND  PRACTICAL  ADVICE  TO 
THOUSANDS. 

It  is  the  belief  of  many  observant  philosophic  persons 
who  are  well  on  their  way  tlirough  life,  that  it*  people 
generally  knew  more  they  would  behave  better,  though 
few,  if  any  of  them,  believe  that  knowledge  and  morality 
are  synonymous  terms.  Acting  on  this  conviction,  the 
following  "Hints  to  those  who  keep  Bank  Accounts'' 
have  been  suggested  by  a  gentleman  well  qualified  by 
general  intelligence  and  long  practical  experience  to  ad- 
vise the  young  and  untaught  of  the  several  matters. 
HINTS  TO  THOSE  WHO  KEEP  BANK   ACCOUNTS. 

1.  If  you  wish  to  open  an  account  with  a  bank,  pro- 


282      ORTON  &  Sadler's  calculator. 

vide  yourself  with  a  proper  introduction.    Well-managed 
banks  do  not  open  accounts  with  strangers. 

2.  Do  not  draw  a  check  unless  you  have  the  money  in 
bank  or  in  your  possession  to  deposit.  Don't  test  the 
courage  or  generosity  of  your  bank  by  presenting,  or  al- 
lowing to  be  presented,  your  check  for  a  larger  sum  than 
your  balance. 

3.  Do  not  draw  a  check  and  send  it  to  a  person  out  of 
the  city,  expecting  to  make  it  good  before  it  can  possibly 
get  back.  Sometimes  telegraphic  advice  is  asked  about 
such  checks. 

4.  Do  not  exchange  checks  with  anybody.  This  is  soon 
discovered  by  your  bank ;  it  does  your  friend  no  good 
and  discredits  you. 

5.  Do  not  give  your  check  to  a  friend  with  the  condi- 
tion that  he  is  not  to  use  it  until  a  certain  time.  He  is 
sure  to  betray  you,  for  obvious  reasons.  Do  not  take  an 
out-of-town  check  from  a  neighbor,  pass  it  through  your 
bank  without  charge,  and  give  him  your  check  for  it. 
You  are  sure  to  get  caught. 

6.  Do  not  give  your  check  to  a  stranger.  This  is  an 
open  door  for  fraud,  and  if  your  bank  loses  through  you, 
it  will  not  feel  kindly  to  you. 

7.  When  you  send  your  checks  out  of  the  city  to  pay 
bills,  write  the  name  and  residence  of  your  pnyee,  thus: 
Pay  to  Jno.  Smith  &  Co.,  of  Boston.  This  will  put  your 
bank  on  its  guard,  if  presented  at  the  counter. 

8.  Don't  commit  the  folly  of  supposing  that,  because 
you  trust  the  bank  with  your  money,  the  bank  ought  to 
trust  you  by  paying  your  overdrafts. 

9.  Don't  suppose  you  can  behave  badly  in  one  bank 
and  stand  well  with  the  others.  You  forget  there  is  a 
Clearing  House. 


BANK   ACCOUNTS.  283 

10.  Don't  quarrel  with  your  bank.  If  you  are  not 
treated  well,  go  somewhere  else,  but  don't  go  and  leave 
your  discount  line  unprotected.  Don't  think  it  unreason- 
able if  your  bank  declines  to  discount  an  accommodation 
note.  Have  a  clear  definition  of  an  accommodation  note 
— in  the  meaning  of  a  bank,  it  is  a  note  for  which  no 
value  has  passed  from  the  endorser  to  the  drawer. 

11.  If  you  want  an  accommodation  note  discounted, 
tell  your  bank  frankly  that  it  is  not,  in  their  definition,  a 
business  note.  If  you  take  a  note  from  a  debtor  with  an 
agreement,  verbal  or  written,  that  it  is  to  be  renewed  in 
whole  or  in  part,  and  if  you  get  that  note  discounted,  and 
then  ask  to  have  a  new  one  discounted  to  take  up  the  old 
one,  tell  your  bank  all  about  it. 

12.  Don't  commit  the  folly  of  saying  that  you  will 
guarantee  the  payment  of  a  note  which  you  have  already 
endorsed, 

13.  Give  your  bank  credit  for  being  intelligent  gener- 
ally and  understanding  its  own  business  particularly. 
It  is  much  better  informed,  probably,  than  you  suppose. 

14.  Don't  try  to  convince  your  bank  that  the  paper  or 
security  which  has  already  been  declined  is  better  than 
the  bank  supposes.     This  is  only  chaff. 

15.  Don't  quarrel  with  a  teller  because  he  does  not  pay 
you  in  money  exactly  as  you  wish.  As  a  rule,  he  does 
the  best  he  can. 

16.  In  all  your  intercourse  with  bank  officers,  treat 
them  with  the  same  courtesy  and  candor  that  you  would 
expect  and  desire  if  the  situations  were  reversed. 

17.  Don't  send  ignorant  and  stupid  messengers  to  bank 
10  transact  your  business. 


284      ORTON  &  Sadler's  calculator. 

\ 

INTEREST-COMMERCIAL  RULE. 

Commercial  Tear.  —  Calculations  based  on  360 
days  to  the  year,  or  30  days  to  the  month. 

SIX    PER    CENT. 

RuLK. — Multiply  the  given  number  of  dollars  hy 
the  number  of  days  of  interest  required ;  divide  the 
product  by  6,  and  point  off  three  figures  from  the 
right. 

Note — If  cents  appear  in  the  principal,  it  will  be 
necessary  to  point  off  five  figures. 

The  result  is  yf^  more  than  the  true  interest  bashed 
on  the  calculation  of  365  days  per  annum. 

To  ascertain  the  true  amount,  it  will  only  be  neces- 
sary to  deduct  ^5  from  the  result  obtained. 

Example.  —  What  is  the  interest  on  $1000  for 
219  days? 

Process.—  1000   X  219  =  219000 
219000  -r-       6  =     36500 

Point  off  three  figures  from  the  right,  gives 
$36.50  interest. 

To  ascertain  the  true  interest  (365  days),  subtract 
Vy,  thus  ;     36.50  -j-  73  ==  50.     $36.50  —  50  =  $36. 

Having  ascertained  the  interest  at  6  per  cent., 
that  for  7,  8,  and  9  per  cent,  is  readily  foimd,  by 
adding  to  it  i,  §,  ^,  etc.,  etc. 

To  find  the  interest  at  any  per  cent.,  divide  the 


INTEREST — COMMERCIAL   RULE.  285 

interest  procured  at  6  per  cent,  by  6,  and  multiply 
the  amount  by  the  required  rate. 

VALUABLE    INTEREST    RULES. 
Basis  Commercial  Year  of  360  days,  or  30  days  per  month. 

4  per  cent. — Multiply  the  principal  by  the  required 
number  of  days,  divide  by  9,  and  point  off. 

5  per  cent. — Multiply  by  the  number  of  days,  and 
divide  by  72. 

6  per  cent.  —  Multiply  by  the  number  of  days, 
divide  by  6,  and  point  off  three  figures  from  the  right. 

8  per  cent. — Multiply  by  the  number  of  days,  and 
divide  by  45. 

9  per  cent. — Multiply  by  the  number  of  days,  divide 
by  4,  and  point  off  three  figures  from  the  right. 

10  per  cent. — Multiply  by  the  number  of  days,  and 
divide  by  36. 

12  per  cent. — Multiply  by  the  number  of  days,  divide 
by  3,  and  point  off  three  figures  from  the  right. 

15  per  cent.-^  Multiply  by  the  number  of  days,  and 
divide  by  24. 

18  per  cen^— Multiply  by  the  number  of  daya, 
divide  by  2,  and  point  off  three  figures  from  the  right. 

20  per  cew/.— Multiply  by  the  number  of  days,  and 
dividxi  by  18. 

iS^The  interest  in  each  case  will  be  in  dollars 
and  c^nlB. 


Rate,  10  per  cent,  per  annum  of  360  days. 
These  Tables  are  arranged  with  a  view  to  supply  a 
want  long  existing  among  the  great  majority  of  busi- 
ness men  and  accountants,  and  are  specially  designed 
to  supersede  the  numerous  high-priced  Interest 
Tables  now  before  the  public. 

CALCULATIONS. 

The  basis  is  at  the  decimal  rate  of  10  per  cent,  per 
annum  on  the  commercial  year  of  360  days,  and  the 


for  which  our  calculations  of  interest  are  made  is 
from  1  to  30  days,  and  from  1  to  12  months,  on 
amounts  ranging  from  $10  to  $10,000,  thereby  meet- 
ing the  wants  of  the  capitalist  as  well  as  those  of  the 
more  moderate  tradesman. 


THE    CALCUIJITIONS 

are  of  the  most  minute  accuracy,  being  carried  out  in 
each  instance  to  the  fraction  of  one  tenth  of  one  mill ; 
286 


INTEREST   TABLES.  287 

hence,  in  gnmming  up  the  araoiint  of  interest,  it  will 
always  be  necessary  to  point  o^  four  figures  from  the 
left. 

When  amounts  from  $1  to  $10  appear  in  the 
principal,  the  interest  is  obtained  by  taking  the  sum 
in  the  interest  column  opposite  the  required  amounts 
from  $10  to  $90,  and  point  off  five  figures  from  the 
left. 

CENTS    IN    THE    PRINCIPAL. 

When  there  are  cents  in  the  principal  in  excess 
of  50,  add  $1.00,  if  less,  reject  them  ;  in  the  calcula- 
tion of  interest  this  is  in  accordance  with  usual  cus- 
tom. It  is  also  customary  when  the  fraction  of 
interest  is  5  mills,  or  in  excess,  to  add  one  cent;  when 
less,  drop  it. 

TO    ASCERTAIN    INTEREST 

on  the  basis  of  365  days  per  annum. 
Subtract  i^^  from  the  results  obtained  by  tlie  calcu- 
lations on  the  basis  of  360  days  per  annum,  which  is 
equivalent  to  a  reduction  of  IJ  cents  for  every  dollar 
of  interest. 

TO    ASCERTAIN    INTEREST 

at  any  rate  other  than  10  per  cent. 
Multiply  the  amount  of  interest  obtained  from  the 
Tables  by  the  required  rate,  and  point  off  five  figures 
from  the  left. 

INSTRUCTION. 

It  will  only  be  necessary  to  trace  the  work  of  the 
following  examples  to  enable  any  one  to  become  ex* 
pert  in  the 


288      ORTON  &  Sadler's  calculator. 

USE   OF   THE   TABLES. 

In  calculating  interest,  refer  to  the  Table  containing 
at  its  head  the  number  of  days  or  months  for  which 
interest  is  required,  and  opposite  the  principal  in  the 
column  of  dollars  will  be  found  the  interest  in  dollars, 
cents,  mills  and  tenths  of  mills. 

Example  I. — Ascertain    the    interest    on    $3570 
for  28  days  at  10  per  cent. 
See  Table,  page  204. 

Dollar  Colnmn.  Interest. 

$3000  $23.3333 

500  3.8889 

70  .5444 


$3570  $27.7666 

Interest  at  10  per  cent.,  28  day*. 

To  find  the  Interest  on  above  amount  at  6  per 
cent: 

Interest  at  10  per  cent.,        $27.7666 
Multiply  by  required  rate,  6  per  cent. 


$16.65996 

Point  off  five  figures  from  the  left  and  you  will 
have  the  interest  at  6  per  cent.,  $16.66. 

OR,  THUS  : 

By  removing  the  decimal  point  one  place  to  the  left 
we  have  the  interest  at  1  per  cent. ;  hence,  by  simply 
m^zltiplying  by  the  required  rate,  the  product  will  be 
the  desired  interest. 


INTEREST   TABLES.  289 

Examiile   IL — Ascertain   the    interest  on    $2007 
for  7  months,  17  days,  at  10  per  cent, 
^ee  Table,  page  206,  7  months. 

Dollar  Column.  Interest. 

$2000  $116.6667 

7       ($70,— $4.0833)  .4083 

See  Table,  page  202,  17  days. 

Dollar  Column. 

$2000  9.4444 

7       ($70.--     .3306)  .0330 


$2007  $126.5524 

Interest  7  months,  17  days,  10  per  cent. 

To  find  the  interest  on  above  amount  at  7   per 
cent : 

Interest  at  10  per  cent.,        $126.5524 
Multiply  by  required  rate,  7  per  cent. 


$88.58668 


Point  off  five  figures  from  the  left,  and  we  have  the 
interest  at  7  per  cent.,  $88.59. 

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58??5 

$666.66 
600  00 
533.33 
466.66 
400.00 

333.33 
266.6ei 
200  00 
133.33 
66.66 

60.00 
53.33 
46.66 
40.00 
33.33 

26.66 
20.00 
13.33 
6.66 

2.66 

2.00 

1.33 

.66 

7  months, 
210°days. 

S?85?38 

5??8?o?? 

8?HS?SS 

?S8I^S? 

85S585 

S585S5 

$583.33 
525.00 
466.66 
408.33 
350.00 

291.66 
233.33 
175.00 
116.66 
68.33 

52.50 
46.06 
40.83 
35.00 
29.16 

23.33 

17.50 

11.66 

5.83 

tartan  tN* 

2.33 

1.75 

1.16 

.58 

6  months, 
180°days. 

888S8 

88888 

88888 

8888 

88888 

8888 

S|8|8 

250.00 
200  00 
150.00 
100.00 
50.00 

45.00 
4000 
35.00 
30  00 
25.00 

20.00 
15  00 
10.00 
5.00 

S888S 

"•t  Ti^  CO  CO  ci 

2.00 

1.50 

1.00 

.50 

gsf 
a  g 

U3      rH 

5S?558 

??5s8eS5 

8S5?58S? 

58??5 

sn^su 

?5853& 

$416.66 
375.00 
333.33 
291.66 
250.00 

208.33 
166.66 
125.00 
83.33 
41.66 

37  50 
33.33 
29.16 
25.00 
20.83 

«o*  (>i  00*  Tji 

3.75 
3.33 
2.91 
2.50 
2.08 

1.66 

1.25 

.83 

.41 

4  months, 

or 
120  days. 

?38lo?2g 

5S58':o'c? 

8?o??S& 

?28?o?? 

S^?^SIB 

?585?3 

^s's^s 

166.66 
133.33 
100.00 
66.66 
33.33 

30-00 
26.66 
23.33 
20.00 
16.66 

1333 

10.00 

6.66 

3.33 

3.00 
2.66 
2.33 
2.00 
1.66 

1.33 

1.00 

.66 

.33 

3  months, 
90  days. 

S88SS 

88838 

88888 

8888 

88888 

8888 

$250.00 
225.00 
200.00 
175.00 
150.00 

125.00 
100.00 
7500 
50.00 
25.00 

22.50 
20.00 
17.50 
15.00 
12.50 

10.00 
7.50 
5.00 
2.50 

2.25 
2.00 
1.75 
1.50 
1.25 

1.00 
.75 
.50 
.26 

2  months, 

or 
60  days. 

•coS??SS 

S5'c5S??!o 

8??5S?? 

^SU^ 

S^^S^ 

?o8?35 

$166.66 
150.00 
133.33 
116.66 
100.00 

83.33 
66.66 
60.00 
33  33 
16.66 

o  JO  — *  d  00 

99*1 
88  8 
00*9 
99*9 

1.50 
1.33 
1.16 
1.00 
.83 

SgS5S 

1  month, 

or 
30  days. 

??85??8 

5??85?? 

§l=??SJo 

?58&S? 

g5??S5 

g585?3 

$83.33 
75.00 
66.66 
68.33 

60.00 

41.66 
33.33 
25.00 
16.66 
8.33 

3.33 

2.50 

1.66 

.83 

^.8Sg5! 

??.{nSS 

1 

0008 
0006 
OOOOI 

iiiii 

imm 

8SSS 

oaooocoio 

SS22 

293 


294 


ORTON    &   SADLER  8    CALCULATOR. 


Orton  &  Sadler's  Interest  Tables. 
Rate  10  per  cent,  360  days  per  annum. 


9  montlis, 

10  montlis, 

11  months, 

12  months, 

Dolls. 

or 

or 

or 

or 

Dolls, 

270  days. 

300  days. 

330  days. 

360  days. 

10000 

STnO.OO 

00 

^^■■iWM 

:53 

Srll6  66 

67 

$1000.00:00  i 

10000 

9000 

675.00 

00 

7511.00 

00 

825.00 

00 

900.00 

00 

9000 

8000 

600.00 

0.) 

6GG.GG 

G7 

733.33 

33 

800.00 

00 

8000 

7000 

52.5.00 

00 

58:',.:53 

33 

641.66 

67 

700.<H) 

00 

m 

6000 

450.00 

uo 

600.U0 

00 

650.00 

00  • 

€00  00 

00 

5000 

375.00 

00 

416.60 

67 

458.33 

33 

600.00 

00 

5000 

4000 

300.00 

00 

333.33 

33 

36(566 

67 

400t»0 

00 

4000 

3000 

225.00 

00 

250.00 

00 

275.00 

00 

300.00 

(X) 

3000 

2000 

150.00 

00 

106  GO 

67 

183.33 

33 

200  00 

00 

2000 

1000 

75.00 

00 

83.33 

33 

91.66 

67 

100.00 

(iO 

1000 

900 

67.50 

00 

75.00 

00 

82.50 

00 

90.00 

00 

900 

1800 

60.00 

00 

66  66 

67 

73.33 

3;i 

80  00 

00 

700 

52.50 

00 

58.33 

33 

64.16 

67 

70.00 

00 

700 

600 

45.00 

00 

50  00 

00 

65.00 

00 

60.00 

00 

600 

500 

37.50 

00 

41.66 

67 

45.83 

33 

60.0U 

00 

500 

400 

30  00 

00 

33.33 

33 

36.66 

67 

40.C0 

00 

400 

300 

22.50 

00 

25.(;o 

00 

27.50 

00 

80.0U 

00 

300 

200 

15.0;i 

00 

16.66 

t>7 

18.33 

33 

20.00 

00 

200 

100 

7.50 

00 

8.33 

33 

0.16 

67 

10.10 

00 

100 

90 

6  75 

00 

7.50 

00 

8.25 

00 

O.f'O 

00 

90 

80 

0.0) 

00 

6.  d 

67 

7.33 

33 

8.00 

00 

80 

70 

5.25 

00 

6.83 

33 

6.41 

67 

7.00 

00 

70 

60 

4.50 

m 

5.00 ;  00 

6..50 

00 

6.00 

00 

60 

50 

S.75 

0. 

4.16;  67 

4.58 

33 

6.00 

00 

50 

40 

8.00 

00 

8.33 

33 

3.66 

67 

400 

00 

40 

30 

i-'.25 

•!0 

2.50 

(M) 

2.75 

00 

3.00 

00 

30 

20 

1.50 

00 

1.66 

67 

1.83 

33 

2.00 

00 

20 

ID 

.75 

00 

.83 

33 

.Ul 

67 

1.00 

00 

10 

COMPOUND   INTEREST. 


295 


COMPOUND  INTEREST  TABLE, 

SlwAoiiig  the   amount  of  $1.00   at  Compound   Interestf  from  1  to 
20   year*.     Hate   5    to   10    -per   cent. 


Trars. 

5  Per  Cent. 

6  Percent. 

7  Per  Cent. 

10  Percent. 

1 

1.050000 

1.060000 

1.070000 

1.100000 

2 

1.102000 

1.123600 

1.144900 

1.210000 

3 

1.157G25 

1.191016 

1225043 

1.331000 

4 

1.215506 

1.262477 

1.310796 

1.464100 

l> 

1.2762B2 

1.338226 

1.402552 

1.610510 

6 

1.340096 

1.418519 

1.500730 

1.771561 

••7 

1.407100 

1.503030 

1.605781 

1.948717 

8 

1.477455 

1.593848 

1.71H186 

2.143589 

9 

1.551328 

1.689479 

1.838459 

2.357948 

10 

1.628895 

1.790848 

1.967151 

2.593742 

11 

1.710339 

1.898299 

2.104852 

2.853117 

12 

1.795856 

2.012196 

2.25-il9J 

3.138428 

13 

1.885619 

2.132928 

2.409845 

3.4."227l 

14 

1.979932 

2.200904 

2.578534 

3.79749S" 

.  1» 

2.078928 

2.396558 

2.759031 

4.177248 

16 

2.182S75 

2.540352 

2.952164 

4.594973 

17 

2.292018 

2.692773 

3.158815 

5.054470 

18 

2.400019 

2.854339 

3.379912 

5.559917 

19 

2.520950 

3.025599 

3.616527 

6.115909 

20 

2.653298 

3.207135 

3.869684 

6.727500 

N.  B.  In  the  calculations  of  Compound  Interest, 
much  labor  will  be  saved  by  use  of  the  above  Table. 

RuLr:. — Refer  to  the  Ttible,  ascertain  the  amount 
of  $1.00  for  (he  giv^n  time  at  the  specified  rate,  and 
multiply  same  hy  the  principal. 


296 


ORTON    <fe    BADLERS    CALCULATOR. 


Example. — What  will  be  the  amount  of  $G00  fo? 
15  years,  at  6  per  cent.  Compound  Interest  ? 

Process. — See  Table.  Amount  $1.  for  15  years 
at  6  per  cent $2.396558 


Amount., 


GOO 


Am't  of  $600  for  15  yrs.  at  6  per  cent.  .$143'i. 934800 

TO   FIND   THE    COMPOUND    INTEREST, 
ime  and  Rate  heinfj  given. 

Rule. — Subtract  the  principal  invested  from  the 
amount. 

Example. — What  is   the  Compound   Interest   on 
$6,00  for  15  years,  at  6  per  cent.  ? 

From  the  above  Example,  $600,  15  years,  6  per 
cent.,  Comp.  Int.,  we  have  the  amount $1437.93 

Principal  invested 600. 

Compound  Interest $837.93 

Process.  —  $000   invested  at  Compound   Interest, 
for  15  year"  at  6  per  cent.,  will  amount  to  $1437.93. 

TABLE 
Showing  in  how  many  Ykars  a  given  principal  will 

DOUBLE  ITSELF. 


At  Compound  Interest. 

Rate. 

Inteuest. 

Compotindecl 

Componnd('d 

Yearly. 

Halt-Yearly. 

43^ 

22.22 

15.748 

15.570 

6 

20.00 

14.207 

14.036 

6 

lO.fJT 

11.896 

11.725 

7 

14.29 

10.245 

10.075 

8 

12.o0 

9.006 

8.837 

9 

11.11 

8.043 

7.874 

10 

10.00 

7.273 

7.+ 

INTEREST   RATES. 


297 


TABLE  OF  INTEREST  RATES  FOR  THE  U.  8. 

Penalties  for  Usury^  and  Statttfe  Limitations. 


STATES 

AND 

TERRITORIES. 


Alabama 

Arizona 

Arkansas 

Calir«jrnia 

Colorado 

Connecticut 

Dakota 

Delaware  

District  of  Columbia, 

Florida 

Georgia 

Idaho 

Illinois 

Indiana....... 

Iowa 

Kansas  

Kentucky 

Louisiana 

Maine 

Iklarj'land 

Massiichusetts  

Michigan 

Minnesota 

Missis-sippi 

Missouri 

Montana 

Nebnj^ka 

Kevada 

New  Hampshire 

New  Jersey 

New  Mexico 

New  York .*. 

Noilh  Carolina 

Ohio.. 

Oregon 

Pennsylvania 

BUode  Island 


illl 


10 
G 

10 

10 
6 
7-10 
6 
6 
8 
7 

10 
6 
6 
6 

7 

6 

5 

6 

6 

G 

7 

7 

6 

G 

10 

10 

10 


Any 


6 
Any 
0 
10 
Any 
10 
Any 
10 
10 
10 

12 

10 

8 

Any 

G 

Any 

10 

12 

10 

10 
Any 

12 
Any 


12 
Any 

Any 


Penalties  fob  Usury. 


Forfeiture  of  entire  interest. 


Forfeiture  of  entire  interest. 


Forfeiture  of  the  principal... 
Forfeiture  of  entir^j  interest. 


Forfeiture  of  excess.. 


Forfeiture  of  entire  interest. 

Forfeiture  of  excess 

Forfeiture  of  entire  interest ; 

10  per  ct  of  it  to  sch.  fund. 

Forfeiture  of  entire  interest. 

Forfeiture  of  excess 

Forfeiture  of  entire  interest. 


Statute 
LiuiitatiQua. 


Forfeiture  of  excess.. 
Forfeiture  of  excess.. 


Forfeiture  of  entire  interest. 


Forfeiture  of  entire  interest. 


Forfeiture  of  three  times  the 

excess  and  costs 

Forfeiture  of  entire  interest, 


Forfeiture  of  excess 

Forfeiture  of  entire  interest. 
Forfeiture  of  excess 


5 
6 

5 
3 

3 

6 
3 

64:10 
6 
6 


Yrs, 
6 


7i-21' 
4' 
4/ 
17 
15 
6^20 

5 

8 


6 
3 
6 
6 
G 
6(?) 


9 

20' 

10' 
5 
7 
5. 

6 
3 
6x20i 
«* 
« 
6 
10 


3 
15 

6 
5,2:0 

6 


298      ORTON  &  Sadler's  calculator. 


TABLE  OF  INTEREST  HATES— Continued, 


STATES 

AND 

TERRITORIES. 


fioiith  Carolina.. 
Tennessee 


Toxas 

Utah  

Vermont... 
Virginia... 


Waahinj^on 

West  Virginia  . 
Wisconsin 


W    V 

!i^ 

:s  S 

^^- 

^^ 

x4 

-< 

Per 

Per 

Ct. 

Ct. 

7 

Any 

G 

10 

8 

Any 

10 

Any 

6 

G 

6 

12 

10 

Any 

(i 

b 

7 

10 

Penalties  fob  Usuky. 


Forfeiture    of  excess;    fine 
and  imprisonment 


Forfeiture  of  excess 

Forfeiture  of  excess,  in  ac- 
tion of  equihj 


Forfeiture  of  excess 

Forfeiture  of  entire  interest. 


Statute 
LiniiUilioD* 


Yrs. 


G,14 
5,20 
5,*  20 


Notes. — The  legal  rate  of  interest  in  Canada,  Nova 
8ootia,  and  Ireland,  is  6  per  cent.;  England  and 
France,  5  per  cent. 

When  the  rate  is  not  specified,  the  legal  rate  is 
a' ways  understood  and  so  allowed  by  the  courts. 

Debts  of  all  kinds  draw  interest  from  the  time  ihey 
become  due,  but  not  before,  unless  specified. 


WEALTH   FROM   SAVINGS, 


299 


HOW  TO  OBTAIN  WEALTH. 

Table  showing  the  net  amount  of  earnings  of  One  Cent  to  Twenty- 
five  Dollars  per  Day  for  Ten  Years  of  313  working  days,  without 
Interest,  and  with  interest  at  6,  7,  and  8  per  cent.,  compounded 
each  Six  Months. 


S'lvings 

Without 

With  interest 
at  G 

With  interest 
at? 

With  inlir(.':»t 
at  « 

per  day. 

interest. 

pur  cent. 

per  cent. 

P'.*r  cent. 

1 

$31    13 

$42  05 

$44  26 

$46  CO 

2 

62  26 

84  10 

88  62 

93  21 

3 

93  39 

126  16 

132  77 

139  81 

4 

r_'4  62 

168  21 

177  03 

186  41 

5 

156  60 

210  26 

222  29 

233  01 

C 

187  80 

252  31 

265  66 

279  62 

7 

219  10 

294  36 

309  80 

326  22 

8 

250  40 

336  62 

354  06 

372  82 

9 

281  70 

378  47 

o98  22 

419  42 

10 

313  00 

420  62 

442  58 

46Q  03 

15 

469  60 

630  78 

666  87 

699  04 

20 

626  00 

841  04 

SSo  15 

932  05 

25 

782  50 

1,051  30 

1,111  44 

1,165  07 

30 

939  00 

1,261  66 

1,327  73 

1,398  03 

40 

1,252  0) 

1,682  09 

1,770  31 

1,864  11 

60 

1,565  00 

2,102  61 

2,212  89 

2,330  13 

60 

1,878  oa 

2,523  13 

2,655  46 

2,796  16 

70 

2,191  00 

2,943  65 

3,(;98  04 

3,262  19 

80 

2,504  00 

3,364  17 

3,540  02 

3,728  22 

90 

2,817  00 

3,784  69 

3,982  19 

4,194  24 

$1  00 

3,130  00 

4,205  21 

4,425  77 

4,660  27 

2  00 

6,260  00 

8,410  43 

8,861  64 

9,320  64 

3  00 

9,390  00 

12,615  64 

13,277  31 

13,980  81 

4  00 

12,520  00 

16,820  So 

17,703  08 

18,641  08 

6  00 

15,650  00 

21,026  07 

22,228  85 

23,301  -35 

6  00 

18,750  00 

25,251  28 

26,654  32 

27,961  62 

7  00 

21,i)10  00 

29,436  50 

30,980  39 

32,621  89 

8  00 

25,040  01) 

33,642  71 

35,400  16 

37,282  14 

9  00 

28,170  00 

37,846  92 

39,821  93 

41,942  42 

10  00 

31,300  00 

42,052  14 

44,257  70 

46,602  69 

15  00 

46,950  00 

63,078  20 

66,668  65 

69,904  04 

20  00 

6l',600  00 

84,104  27 

88,515  40 

93,205  39 

25  00 

78,250  00 

105,030  00 

111,144  00 

116,507  CO 

From  the  above  Table  it  can  readily  be  observed  wliy 
"Fortunes  are  Spent  by  Trifles,"  and  the  advantage  in 
saving,  if  one  desires  to  obtain  a  competency.  This  Table 
is  worUiy  t!ie  careful  utteiitiou  >f  our  young  inc-i  whc 
desire  success  in  life. 


WAGES-VALUE   OF  TIME. 

For  Days  and  Hours,  at  Stated  Rates  Per  Week 


Rat» 

$^1  3K 

$44>^ 

$5 

5M 

$6|63^ 

♦7 

VA 

$8 

$9 

10 

11 

12 

^1 

5 

6 

7 

8 

8 

9 

.10 

.11 

.12 

.13 

.13     .15 

.17 

.18 

.20 

S'-i 

.10 

.12 

.13 

.15 

.17 

.18 

.20 

.22 

.23 

.25 

.27     .30 

.33 

.37 

.40 

ri 

.15 

.18 

.20 

.23 

.25 

.28 

.30 

.33 

35 

.38 

.40     .45 

.50 

.55 

.00 

.'20 

.23 

.27 

.30 

.33 

.37 

.40 

.43 

.47 

.50 

.53     .60 

.67 

.73 

.80 

5 

.'25 

.29 

.33 

.38 

.42 

.46 

.50 

.54 

.58 

.63 

.67 

.75 

.83 

.92 

1.00 

6 

.ao 

.35 

.40 

.45 

.50 

.55 

.60 

.60 

.70 

.75 

.80 

.90 

l.{!0 

1.10 

1.20 

7 

.35 

.41 

.47 

.53 

.58 

.64 

.70 

.76 

.82 

.88 

.93 

1.05 

1.17 

1.28 

1.40 

8 

.40 

.47 

.53 

.60 

.67 

.73 

.80 

.87 

.93 

1.00 

1.07 

1.20 

1.33 

1.47 

1.60 

V 

.45 

.53 

.GO 

.68 

.75 

.83 

.90 

.98 

1.05 

1.13 

1.20 

1.35 

1.50 

1.65 

1.80 

1^ 

.50 

.58 

.07 

.75 

.83 

.92 

1.00 

1.08 

1.17 

1.25 

1.33 

1.50 

1.67 

1,83 

2.00 

1.00 

1.17 

1.33 

1.50 

1.67 

1.83 

2.00 

2.17 

•3.33 

2.50 

2.67 

3.00 

3.3.1 

i  67 

4.00 

1.50 

1.75 

2.00 

2.25 

2.50 

2.75 

3.00 

3.25 

3.50 

3.75 

4.00 

4.50 

o.t!0 

5.50 

6.00 

4 

2.00 

2.33 

2.67 

3.00 

3.33 

3.67 

4.00 

4.33 

4.67 

5.00 

5.33 

6.00 

6.67 

7. .33 

8.00 

5 

2.50 

2.92 

3.33 

3.75 

4.17 

4.58 

5.00 

5.42 1 5.83 

6.25 

0.67 

7.50 

8.33 

.).ll 

$10. 

For  Day  Sy  at  Stated  Bates  Per  Month. 


Rat^. 

$14 

$15 

$16 

»n 

$18 

m 

$20 

$21 

$22  $23  $24 

$25 

^  3 

.54       .58 

.62 

.65 

.69 

.73 

.77 

.81 

.85 

.88 

.92 

.96 

1.08    1.15 

1.23 

1.31 

1.38 

1.46 

1.54 

1.62 

1.69 

1.77 

1.85 

1.92 

1.62    1.73 

1.85 

1.96 

2.08 

2.19 

2.31 

2.42 

2.54 

2.65 

2.77 

2.88 

4 

2.15 

2.31 

2.46 

2.62 

2.77 

2.92 

3.08 

3.23 

3.38 

3.54 

3.69 

3.85 

6 

2.69 

2.88 

3.08 

3.27 

3.46 

3.65 

3.85 

4.04 

4.23 

4.42 

4.62 

4.81 

e 

3.23 

3.46 

3.69 

3.92 

4.15 

4.38 

4.62 

4.85 

5.08 

5.31 

5.54 

5.77 

7 

3.77 

4.04 

4.31 

4.58 

4.85 

6.12 

5.38 

5.65 

5.92 

6.19 

6.46 

6.73 

8 

4.31 

4.62 

4.92 

5.23 

5.54 

5.85 

0.15 

6.46 

6.77 

7.08 

7.38 

7.69 

9 

4.85 

5.19 

5.54 

5.88 

6.23 

6.58 

6.92 

7.2T 

7.62 

7.96 

8.31 

8.65 

10 

5.38 

5.77 

6.15 

6.54 

6.92 

7.31 

7.69 

8.08 

8.46 

8.85 

9.23 

9.62 

11 

6.92 

6.35 

6.77 

7.19 

7.62 

8.04 

8.46 

8.8^ 

9.31 

9.73110.15 

10.58 

12 

6.46 

6.92 

7.38 

7.85 

8.31 

8.77 

9.23 

9.69 

10.15 

10.02 

11.08 

11.54 

13 

7.00 

7.50 

8.00 

8.50 

9.00 

9.50 

10.00 

10.50 

11.00 

11.50 

12.00 

12.60 

14 

7.54 

8.08 

8.62 

9.15 

9.69 

10.23 

10.77 

11.31 

11.85 

12.38 

12.92 

13.46 

n 

•8.08 

8.65 

9.23 

9.81 

10.38 

10.96 

11.54 

12.12 

12.69 

13.27 

13.85 

14.42 

16 

•  8.62 

9.23 

9.85 

10.46 

11.08 

11.69 

12.31 

12.92 

13.54 

14.15 

14.77 

15.38 

17 

9.15 

9.81 

10.46 

11.12 

11.77 

12.42 

13.08 

1 3.73114.38 

16.04 

15.69 

16.35 

18 

9.69 

10.38 

11.08 

11.77 

12.46 

13.15 

13.85 

14.54 

15.23 

16.92 

16.62 

17.31 

19 

10.23 

10.96 

11.69 

12.42 

13.15 

13.88 

14.62 

15.35 

16.08 

16.81 

17.54 

18.27 

20 

10.77  111.54 

12.31 

13.08 

13.85 

14.62 

15.38 

16.15 

16.92 

17.69 

18.46 

19.23 

21 

11.31 

12.12 

12.92 

13.73 

14.54 

16.35 

16.15 

16.96 

17.77 

18.58 

19.38 120.19 

22 

11.85 

12.69 

13.54 

14.38 

15.23 

16.08 

16,92  17.77 

18.62  i  19.46 

20.31:21.15 

23 

12.38 

13.27 

14.15 

15.04 

15.92  16.81 

17.69118.58 

19.46 

20.35 

21.23122.12 

24 

12.92 

13.85 

14.77 

15.69 

16.62  17.54  18.46 119.38 '20.31 

21.23 

22.15  j  23.08 

25 

13.46 

14.42 

15.38 

16.35 

17.31  18.27^19.23 1 20.19!  21.15 

22.12 

23.08 1 24.04 

26 

14.00 

15.00 

16.00 

17.00 

I8.00!  19.00  20.00 121. 00 ; 22.UO  123.00 i 24.00 125.00 

500 


BEADY    EECKONINQ.  301 

VALUABLE  TABLES 

For  the  Merchant,  Farmer^  and  Purchaser,  showing  at  sight  the 

Value  of  Articles  Sold  by  the  Pound,  Dozen,  Yard,  or 

Piece,  as  Groceries,  Produce,  Dry  Goods,  etc. 

These  Tables  embody  nearly  all  of  the  practical  features 
comprised  in  publications  devoted  exclusively  to  the  sub- 
ject of  Ready  Reckoning,  for  which  prices  are  asked 
nearly  equal  to  the  cost  of  this  entire  work.  They  will 
be  found  invaluable  in  ascertaining  the  value  of  articles 
usually  sold  by  the  Business  Trader  and  Farmer  and  con- 
sumed in  families. 

APPLICATION  OF  THE  TABLES. 

The  outside  perpendicular  columns  to  the  left  and  right 
show  the  price  of  the  article,  and  the  upper  and  lower 
lines  the  quavtUy. 

When  advisable  in  securing  results  the  working  can  be 
reversed. 

Example.— What  will  14  lbs.  of  Coffee  cost  at  29  cts. 
per  pound?  See  price  in  column  to  the  left,  29,  follow 
the  finger  along  the  line  until  the  sum  under  the  column 
14  is  reached,  and  you  have  the  amount,  $4.06. 

Note  I. — When  the  price  or  quantity  required  is  not  shown  in  the 
tables,  reduce  the  number  of  either  or  both  to  such  amounts  aa  are 
shown  in  the  extremes;  ascertain  the  product  from  the  table,  and 
multiply  by  the  number  or  numbers  used  as  a  divisor  in  reduction 
of  the  orlgiual  amounts. 

Example. — ^What  will  450  bushels  of  screenings  cost 
at  23  cts.  per  bushel?  As  450  is  not  contained  in  the 
table,  we  reduce  it  to  45  by  dividing  by  10.  Referring 
to  45  in  column  ^t  the  left,  and  on  tho  same  line  under 
the  head  number  23,  we  have  10.35,  the  cost  of  45  bushels. 
10.35  X  10  --  $103.50,  the  cost  of  450  bushels. 

Note  TI.— "VThen  the  price  or  quantity  is  a  fractionnl  part  of  a  whole 
number,  ascertain  the  amount  from  U)e  tnble  by  URinp:  the  nnmorator 
fts  a  whole  number,  aud  then  divide  by  the  duaominutor,  adding  lbs 
result  to  the  product  already  obtulued. 

26 


TABLE  OP  EBADY  CALCULATIONS. 

See  page  301 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

12 

24 

36 

48 

60 

72 

84 

90 

108 

120 

132 

144 

156 

168 

180 

13 

2Q 

39 

52 

(^0 

78 

91 

104 

117 

130 

M3 

156 

169 

182 

195 

14 

28 

42 

66 

70 

84 

98 

112 

126 

140 

154 

168 

182 

196 

210 

15 

30 

45 

60 

75 

90 

105 

120 

135 

150 

165 

180 

195 

210 

225 

16 

32 

48 

64 

80 

96 

112 

128 

144 

160 

176 

192 

208 

224 

240 

17 

34 

51 

68 

So 

102 

119 

136 

153 

170 

187 

204 

221 

238 

255 

18 

36 

64 

72 

90 

108 

126 

144 

162 

180 

198 

216 

234 

252 

270 

19 

38 

67 

76 

95 

114 

133 

152 

171 

190 

209 

228 

247 

266 

285 

20 

40 

00 

80 

100 

120 

140 

160 

180 

200 

220 

240 

260 

280 

300 

21 

42 

63 

84 

105 

126 

147 

168 

189 

210 

231 

252 

273 

294 

315 

22 

44 

66 

88 

110 

132 

154 

176 

198 

220 

242 

264 

286 

308 

330 

23 

46 

69 

92 

115 

138 

161 

184 

207 

230 

253 

276 

299 

322 

345 

24 

48 

72 

96 

120 

H4 

168 

192 

216 

240 

264 

288 

312 

336 

360 

25 

50 

75 

100 

125 

150 

175 

200 

225 

250 

275 

300 

325 

350 

375 

26 

52 

78 

104 

130 

156 

182 

208 

234 

260 

286 

312 

338 

364 

390 

27 

54 

81 

108 

135 

162 

189 

216 

243 

270 

297 

324 

351 

378 

405 

28 

56 

84 

112 

140 

168 

196 

224 

252 

280 

308 

336 

364 

392 

420 

29 

58 

B>7 

116 

145 

174 

203 

232 

261 

290 

319 

348 

377 

406 

435 

30 

GO 

90 

120 

150 

180 

210 

240 

270 

300 

330 

360 

390 

420 

450 

31 

02 

93 

124 

\bo 

186 

217 

248 

279 

310 

341 

372 

403 

434 

465 

32 

64 

96 

128 

160 

192 

224 

256 

288 

320 

352 

384 

416 

448 

480 

33 

m 

99 

132 

165 

198 

231 

264 

297 

330 

363 

396 

429 

462 

495 

34 

Q^ 

102 

136 

170 

204 

238 

272 

306 

340 

374 

408 

442 

476 

510 

85 

70 

105 

140 

175 

210 

2-15 

280 

315 

350 

385 

420 

4bd 

490 

625 

36 

72 

108 

144 

180 

216 

252 

288 

324 

360 

396 

432 

468 

504 

540 

37 

74 

111 

148 

185 

222 

259 

296 

333 

370 

407 

444 

481 

618 

655 

38 

76 

114 

152 

190 

228 

2(36 

304 

342 

380 

418 

4b^ 

494 

532 

670 

39 

78 

117 

156 

195 

234 

273 

312 

351 

390 

429 

468 

507 

546 

585 

40 

80 

120 

160 

200 

240 

280 

320 

360 

400 

440 

480 

520 

560 

600 

41 

82 

123 

164 

205 

246 

287 

328 

369 

410 

451 

492 

533 

574 

615 

42 

84 

126 

168 

210 

252 

204 

336 

378 

420 

462 

5  (.'4 

546 

588 

630 

43 

^Q 

l-:9 

172 

215 

258 

301 

344 

387 

430 

473 

516 

559 

602 

645 

44 

S^ 

132 

176 

220 

264 

308 

352 

396 

440 

484 

528 

572 

Q\Q 

660 

45 

00 

135 

180 

225 

270 

315 

360 

405 

450 

495 

540 

b^5 

630 

675 

46 

<J2 

138 

184 

230 

276 

322 

368 

414 

4  GO 

506 

552 

598 

644 

690 

47 

94 

HI 

188 

235 

282 

32i3 

376 

423 

470 

517 

564 

611 

658 

705 

48 

1)6 

144 

192 

240 

288 

336 

384 

432 

480 

528 

576 

624 

672 

720 

49 

98 

147 

196 

245 

294 

343 

392 

441 

490 

539 

588 

637 

(\m 

735 

60 

100 

IGO 

200 

250 

300 

350 

400 

450 

500 

550 

600 

650 

700 

750 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

302 


TABLE  OP  EEATY  CALCULATIONS. 

See  page  301 


16 

n 

18 

19 

20 

21 

22 

23 

24 

25 

192 

204 

216 

228 

240 

252 

264 

276 

288 

300 

12 

208 

221 

234 

247 

260 

273 

286 

299 

312 

325 

13 

224 

238 

252 

266 

280 

294 

308 

322 

336 

350 

14 

240 

255 

27(/ 

285 

300 

315 

330 

345 

360 

375 

15 

256 

272 

288 

304 

320 

336 

352 

368 

384 

400 

13 

272 

289 

30t 

323 

340 

357 

374 

391 

408 

425 

1/ 

288 

300 

324 

342 

360 

378 

396 

414 

4;';2 

450 

13 

304 

323 

342 

361 

380 

399 

418 

437 

456 

475 

19 

320 

340 

360 

380 

400 

420 

440 

460 

480 

600 

20 

336 

357 

378 

399 

420 

441 

462 

483 

504 

525 

21 

352 

374 

396 

418 

440 

462 

484 

506 

528 

550 

22 

c6S 

391 

414 

437 

460 

483 

506 

529 

552 

575 

23 

es4 

408 

432 

456 

4  SO 

504 

528 

552 

576 

600 

24 

400 

425 

450 

475 

500 

525 

650 

575 

600 

625 

25 

416 

442 

468 

494 

520 

646 

672 

598 

624 

650 

23 

432 

459 

4S6 

513 

540 

667 

594 

621 

648 

675 

27 

448 

476 

504 

532 

5G0 

688 

616 

644 

672 

700 

23 

464 

493 

522 

551 

580 

609 

638 

667 

606 

725 

29 

480 

510 

540 

570 

600 

630 

660 

690 

720 

750 

30 

496 

527 

553 

589 

620 

651 

682 

713 

744 

775 

3] 

612 

544 

576 

608 

640 

672 

704 

736 

768 

800 

32 

528 

5G1 

504 

627 

660 

693 

726 

759 

792 

825 

33 

644 

578 

6!2 

646 

6^0 

714 

748 

782 

816 

850 

34 

660 

595 

630 

665 

700 

735 

770 

805 

840 

875 

35 

576 

612 

GIS 

6^4 

720 

756 

792 

828 

864 

900 

33 

692 

629 

666 

703 

740 

777 

814 

851 

888 

925 

37 

608 

646 

6S4 

722 

760 

798 

836 

874 

912 

950 

33 

624 

663 

702 

741 

780 

819 

^o^ 

897 

936 

975 

39 

640 

680 

720 

760 

800 

840 

880 

920 

960 

1000 

40 

656 

697 

738 

779 

820 

861 

902 

943 

98; 

1025 

41 

.72 

714 

756 

798 

840 

882 

924 

966 

1008 

1050 

42 

o38 

731 

774 

817 

«60 

903 

946 

969 

1032 

1075 

43 

7)4 

748 

792 

836. 

880 

924 

96S 

1012 

1056 

1100 

44 

/?0 

765 

810 

855 

900 

945 

990 

1035 

1080 

1125 

45 

736 

782 

828 

874 

920 

966 

1012 

1058 

1104 

1150 

43 

752 

799 

846 

893 

940 

i)S7 

1034 

1081 

1128 

1175 

47 

768 

816 

864 

912 

960 

1008 

1056 

1104 

1152 

1200 

43 

•34 

833 

882 

9;u 

980 

1029 

1078 

1127 

1176 

1225 

49 

800 

850 

900 

950 

1000 

1050 

1100 

1150 

1200 

1250 

50 

IG 

n 

13 

19 

20 

21 

22 

23 

21 

25 

303 


^^^  Work  co^^J^m^  ^^^ 


Frontispiece 4 

Addition 15 

Multi!>lica(ion 26 

Counting  llooni 49 

Division,  or  Boxing  Fractio'is 63 

Percentaj^e  as  Applied  to  liiisincsa 60 

Profit  and  Loss 65 

Commission 77 

Stoclcs  and  Investments 81 

Interest 87 

Banking 104 

Averaging  Accounts 135 

Sterling  Excliango 147 

Marking  Goods , 154 

Ledger  Accounts 164 

Closing  Ledger 176 

Trial  Balances — Detecting  Errors 178 

Measuring  Lumber 183 

Measurement  of  Wood 186 

Round  Timber 189 

Squai'e  Timber 194 

Cisterns  and  Reservoirs 197 

Cask  (Jauging 200 

Measuring  {J rain 203 

Corn  Cribs  and  Contents 207 

Hay  in  tbe  Stack 210 

Weigbt  of  Cattle  by  Measurement 213 

Building 215 

Short  Rules  for  ^lechanics 226 

Avoirdupois  Weight  and  Table 247 

Apothecaries'  Weiglit  and  Table 248 

Cord  Wood  and  Table 249 

Dry  Measure  "      "      250 

Cubic    "  "      "      251 

Li<ini(l  '•         "      "      253 

Surveyino:       "      '.'      254 

Square  Measure 256 

Troy  Weight  a nd  Tables 259 

Books  and  Stationery 2G0 

Mejisurement  of  Time 265 

United  States  Money 274 

Subluesd  with  Baukii 281 

304 

V 


SADL,^- 


s^-^^i::^ 


